Dirichlet series
| L(s) = 1 | − 15·2-s − 3-s + 120·4-s − 7·5-s + 15·6-s + 7-s − 680·8-s − 21·9-s + 105·10-s + 15·11-s − 120·12-s − 13-s − 15·14-s + 7·15-s + 3.06e3·16-s − 6·17-s + 315·18-s − 14·19-s − 840·20-s − 21-s − 225·22-s + 2·23-s + 680·24-s − 18·25-s + 15·26-s + 20·27-s + 120·28-s + ⋯ |
| L(s) = 1 | − 10.6·2-s − 0.577·3-s + 60·4-s − 3.13·5-s + 6.12·6-s + 0.377·7-s − 240.·8-s − 7·9-s + 33.2·10-s + 4.52·11-s − 34.6·12-s − 0.277·13-s − 4.00·14-s + 1.80·15-s + 765·16-s − 1.45·17-s + 74.2·18-s − 3.21·19-s − 187.·20-s − 0.218·21-s − 47.9·22-s + 0.417·23-s + 138.·24-s − 3.59·25-s + 2.94·26-s + 3.84·27-s + 22.6·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 11^{15} \cdot 197^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 11^{15} \cdot 197^{15}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(30\) |
| Conductor: | \(2^{15} \cdot 11^{15} \cdot 197^{15}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.22317\times 10^{23}\) |
| Root analytic conductor: | \(5.88278\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(15\) |
| Selberg data: | \((30,\ 2^{15} \cdot 11^{15} \cdot 197^{15} ,\ ( \ : [1/2]^{15} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{15} \) |
| 11 | \( ( 1 - T )^{15} \) | |
| 197 | \( ( 1 - T )^{15} \) | |
| good | 3 | \( 1 + T + 22 T^{2} + 23 T^{3} + 242 T^{4} + 259 T^{5} + 1780 T^{6} + 1909 T^{7} + 9859 T^{8} + 3478 p T^{9} + 4891 p^{2} T^{10} + 5063 p^{2} T^{11} + 166483 T^{12} + 167425 T^{13} + 554380 T^{14} + 535232 T^{15} + 554380 p T^{16} + 167425 p^{2} T^{17} + 166483 p^{3} T^{18} + 5063 p^{6} T^{19} + 4891 p^{7} T^{20} + 3478 p^{7} T^{21} + 9859 p^{7} T^{22} + 1909 p^{8} T^{23} + 1780 p^{9} T^{24} + 259 p^{10} T^{25} + 242 p^{11} T^{26} + 23 p^{12} T^{27} + 22 p^{13} T^{28} + p^{14} T^{29} + p^{15} T^{30} \) |
| 5 | \( 1 + 7 T + 67 T^{2} + 344 T^{3} + 1987 T^{4} + 1661 p T^{5} + 36576 T^{6} + 26108 p T^{7} + 476557 T^{8} + 1490566 T^{9} + 4696122 T^{10} + 13064862 T^{11} + 36308377 T^{12} + 18113716 p T^{13} + 224732477 T^{14} + 504163539 T^{15} + 224732477 p T^{16} + 18113716 p^{3} T^{17} + 36308377 p^{3} T^{18} + 13064862 p^{4} T^{19} + 4696122 p^{5} T^{20} + 1490566 p^{6} T^{21} + 476557 p^{7} T^{22} + 26108 p^{9} T^{23} + 36576 p^{9} T^{24} + 1661 p^{11} T^{25} + 1987 p^{11} T^{26} + 344 p^{12} T^{27} + 67 p^{13} T^{28} + 7 p^{14} T^{29} + p^{15} T^{30} \) | |
| 7 | \( 1 - T + 68 T^{2} - 64 T^{3} + 310 p T^{4} - 1962 T^{5} + 43327 T^{6} - 38903 T^{7} + 611465 T^{8} - 567530 T^{9} + 6589169 T^{10} - 6518677 T^{11} + 57962460 T^{12} - 60931421 T^{13} + 444274088 T^{14} - 469261603 T^{15} + 444274088 p T^{16} - 60931421 p^{2} T^{17} + 57962460 p^{3} T^{18} - 6518677 p^{4} T^{19} + 6589169 p^{5} T^{20} - 567530 p^{6} T^{21} + 611465 p^{7} T^{22} - 38903 p^{8} T^{23} + 43327 p^{9} T^{24} - 1962 p^{10} T^{25} + 310 p^{12} T^{26} - 64 p^{12} T^{27} + 68 p^{13} T^{28} - p^{14} T^{29} + p^{15} T^{30} \) | |
| 13 | \( 1 + T + 114 T^{2} + 167 T^{3} + 6155 T^{4} + 68 p^{2} T^{5} + 210839 T^{6} + 465492 T^{7} + 5182461 T^{8} + 12968755 T^{9} + 98538070 T^{10} + 272792127 T^{11} + 1546948188 T^{12} + 4612769715 T^{13} + 21497775672 T^{14} + 65170275966 T^{15} + 21497775672 p T^{16} + 4612769715 p^{2} T^{17} + 1546948188 p^{3} T^{18} + 272792127 p^{4} T^{19} + 98538070 p^{5} T^{20} + 12968755 p^{6} T^{21} + 5182461 p^{7} T^{22} + 465492 p^{8} T^{23} + 210839 p^{9} T^{24} + 68 p^{12} T^{25} + 6155 p^{11} T^{26} + 167 p^{12} T^{27} + 114 p^{13} T^{28} + p^{14} T^{29} + p^{15} T^{30} \) | |
| 17 | \( 1 + 6 T + 11 p T^{2} + 1013 T^{3} + 16904 T^{4} + 83135 T^{5} + 983709 T^{6} + 4409208 T^{7} + 41351311 T^{8} + 169290928 T^{9} + 1333853479 T^{10} + 4989927012 T^{11} + 34190398218 T^{12} + 6863738554 p T^{13} + 710452257408 T^{14} + 2202093753647 T^{15} + 710452257408 p T^{16} + 6863738554 p^{3} T^{17} + 34190398218 p^{3} T^{18} + 4989927012 p^{4} T^{19} + 1333853479 p^{5} T^{20} + 169290928 p^{6} T^{21} + 41351311 p^{7} T^{22} + 4409208 p^{8} T^{23} + 983709 p^{9} T^{24} + 83135 p^{10} T^{25} + 16904 p^{11} T^{26} + 1013 p^{12} T^{27} + 11 p^{14} T^{28} + 6 p^{14} T^{29} + p^{15} T^{30} \) | |
| 19 | \( 1 + 14 T + 300 T^{2} + 3331 T^{3} + 40802 T^{4} + 19643 p T^{5} + 3381448 T^{6} + 26136121 T^{7} + 192464650 T^{8} + 1279592917 T^{9} + 8008575587 T^{10} + 46379571175 T^{11} + 252455309476 T^{12} + 1284432579383 T^{13} + 6150472628266 T^{14} + 27623525644412 T^{15} + 6150472628266 p T^{16} + 1284432579383 p^{2} T^{17} + 252455309476 p^{3} T^{18} + 46379571175 p^{4} T^{19} + 8008575587 p^{5} T^{20} + 1279592917 p^{6} T^{21} + 192464650 p^{7} T^{22} + 26136121 p^{8} T^{23} + 3381448 p^{9} T^{24} + 19643 p^{11} T^{25} + 40802 p^{11} T^{26} + 3331 p^{12} T^{27} + 300 p^{13} T^{28} + 14 p^{14} T^{29} + p^{15} T^{30} \) | |
| 23 | \( 1 - 2 T + 200 T^{2} - 548 T^{3} + 20255 T^{4} - 2903 p T^{5} + 1384661 T^{6} - 5028844 T^{7} + 71381331 T^{8} - 267945303 T^{9} + 2924623907 T^{10} - 10828230754 T^{11} + 97792628455 T^{12} - 344806075407 T^{13} + 2704382680802 T^{14} - 8821316744326 T^{15} + 2704382680802 p T^{16} - 344806075407 p^{2} T^{17} + 97792628455 p^{3} T^{18} - 10828230754 p^{4} T^{19} + 2924623907 p^{5} T^{20} - 267945303 p^{6} T^{21} + 71381331 p^{7} T^{22} - 5028844 p^{8} T^{23} + 1384661 p^{9} T^{24} - 2903 p^{11} T^{25} + 20255 p^{11} T^{26} - 548 p^{12} T^{27} + 200 p^{13} T^{28} - 2 p^{14} T^{29} + p^{15} T^{30} \) | |
| 29 | \( 1 - 8 T + 197 T^{2} - 1623 T^{3} + 21190 T^{4} - 162527 T^{5} + 1578278 T^{6} - 11016322 T^{7} + 3065029 p T^{8} - 569819661 T^{9} + 4025482566 T^{10} - 23914208125 T^{11} + 153194839007 T^{12} - 850358823241 T^{13} + 5044407000395 T^{14} - 26321625316655 T^{15} + 5044407000395 p T^{16} - 850358823241 p^{2} T^{17} + 153194839007 p^{3} T^{18} - 23914208125 p^{4} T^{19} + 4025482566 p^{5} T^{20} - 569819661 p^{6} T^{21} + 3065029 p^{8} T^{22} - 11016322 p^{8} T^{23} + 1578278 p^{9} T^{24} - 162527 p^{10} T^{25} + 21190 p^{11} T^{26} - 1623 p^{12} T^{27} + 197 p^{13} T^{28} - 8 p^{14} T^{29} + p^{15} T^{30} \) | |
| 31 | \( 1 + 33 T + 767 T^{2} + 12730 T^{3} + 175361 T^{4} + 2006801 T^{5} + 20069800 T^{6} + 174708034 T^{7} + 1357641410 T^{8} + 9339936641 T^{9} + 57849639321 T^{10} + 319033142320 T^{11} + 1604753748485 T^{12} + 7406997595955 T^{13} + 34367492578132 T^{14} + 173977197618249 T^{15} + 34367492578132 p T^{16} + 7406997595955 p^{2} T^{17} + 1604753748485 p^{3} T^{18} + 319033142320 p^{4} T^{19} + 57849639321 p^{5} T^{20} + 9339936641 p^{6} T^{21} + 1357641410 p^{7} T^{22} + 174708034 p^{8} T^{23} + 20069800 p^{9} T^{24} + 2006801 p^{10} T^{25} + 175361 p^{11} T^{26} + 12730 p^{12} T^{27} + 767 p^{13} T^{28} + 33 p^{14} T^{29} + p^{15} T^{30} \) | |
| 37 | \( 1 + 9 T + 291 T^{2} + 2217 T^{3} + 40860 T^{4} + 271002 T^{5} + 3762578 T^{6} + 22426202 T^{7} + 261883471 T^{8} + 1443710739 T^{9} + 14940300415 T^{10} + 77517978731 T^{11} + 728371198323 T^{12} + 3569280289851 T^{13} + 30857870299103 T^{14} + 141992744647530 T^{15} + 30857870299103 p T^{16} + 3569280289851 p^{2} T^{17} + 728371198323 p^{3} T^{18} + 77517978731 p^{4} T^{19} + 14940300415 p^{5} T^{20} + 1443710739 p^{6} T^{21} + 261883471 p^{7} T^{22} + 22426202 p^{8} T^{23} + 3762578 p^{9} T^{24} + 271002 p^{10} T^{25} + 40860 p^{11} T^{26} + 2217 p^{12} T^{27} + 291 p^{13} T^{28} + 9 p^{14} T^{29} + p^{15} T^{30} \) | |
| 41 | \( 1 + 10 T + 495 T^{2} + 4744 T^{3} + 118038 T^{4} + 1071550 T^{5} + 17992274 T^{6} + 153054587 T^{7} + 1959888778 T^{8} + 15471344061 T^{9} + 161509105092 T^{10} + 1172374593144 T^{11} + 253544990081 p T^{12} + 68742507482830 T^{13} + 531893193401127 T^{14} + 3170089393643564 T^{15} + 531893193401127 p T^{16} + 68742507482830 p^{2} T^{17} + 253544990081 p^{4} T^{18} + 1172374593144 p^{4} T^{19} + 161509105092 p^{5} T^{20} + 15471344061 p^{6} T^{21} + 1959888778 p^{7} T^{22} + 153054587 p^{8} T^{23} + 17992274 p^{9} T^{24} + 1071550 p^{10} T^{25} + 118038 p^{11} T^{26} + 4744 p^{12} T^{27} + 495 p^{13} T^{28} + 10 p^{14} T^{29} + p^{15} T^{30} \) | |
| 43 | \( 1 + 6 T + 333 T^{2} + 2057 T^{3} + 58284 T^{4} + 359217 T^{5} + 6999474 T^{6} + 42077594 T^{7} + 640206954 T^{8} + 3689348541 T^{9} + 46985270627 T^{10} + 256113241724 T^{11} + 2848106203708 T^{12} + 14523350288767 T^{13} + 144884762848305 T^{14} + 683686322732544 T^{15} + 144884762848305 p T^{16} + 14523350288767 p^{2} T^{17} + 2848106203708 p^{3} T^{18} + 256113241724 p^{4} T^{19} + 46985270627 p^{5} T^{20} + 3689348541 p^{6} T^{21} + 640206954 p^{7} T^{22} + 42077594 p^{8} T^{23} + 6999474 p^{9} T^{24} + 359217 p^{10} T^{25} + 58284 p^{11} T^{26} + 2057 p^{12} T^{27} + 333 p^{13} T^{28} + 6 p^{14} T^{29} + p^{15} T^{30} \) | |
| 47 | \( 1 + T + 491 T^{2} + 189 T^{3} + 115827 T^{4} - 32793 T^{5} + 17473139 T^{6} - 16823149 T^{7} + 1897136107 T^{8} - 3023325036 T^{9} + 158547265765 T^{10} - 331420584993 T^{11} + 10663702845360 T^{12} - 25113908924347 T^{13} + 594825476692450 T^{14} - 1378192562833348 T^{15} + 594825476692450 p T^{16} - 25113908924347 p^{2} T^{17} + 10663702845360 p^{3} T^{18} - 331420584993 p^{4} T^{19} + 158547265765 p^{5} T^{20} - 3023325036 p^{6} T^{21} + 1897136107 p^{7} T^{22} - 16823149 p^{8} T^{23} + 17473139 p^{9} T^{24} - 32793 p^{10} T^{25} + 115827 p^{11} T^{26} + 189 p^{12} T^{27} + 491 p^{13} T^{28} + p^{14} T^{29} + p^{15} T^{30} \) | |
| 53 | \( 1 - 6 T + 403 T^{2} - 2425 T^{3} + 86571 T^{4} - 505104 T^{5} + 12796530 T^{6} - 71322379 T^{7} + 1438806213 T^{8} - 7597353605 T^{9} + 129652972920 T^{10} - 644117512870 T^{11} + 9651137221947 T^{12} - 44791304897628 T^{13} + 603774659453395 T^{14} - 2594652437842302 T^{15} + 603774659453395 p T^{16} - 44791304897628 p^{2} T^{17} + 9651137221947 p^{3} T^{18} - 644117512870 p^{4} T^{19} + 129652972920 p^{5} T^{20} - 7597353605 p^{6} T^{21} + 1438806213 p^{7} T^{22} - 71322379 p^{8} T^{23} + 12796530 p^{9} T^{24} - 505104 p^{10} T^{25} + 86571 p^{11} T^{26} - 2425 p^{12} T^{27} + 403 p^{13} T^{28} - 6 p^{14} T^{29} + p^{15} T^{30} \) | |
| 59 | \( 1 + 15 T + 579 T^{2} + 7639 T^{3} + 169120 T^{4} + 2002547 T^{5} + 32745051 T^{6} + 350964306 T^{7} + 4668367919 T^{8} + 45505261918 T^{9} + 516847762290 T^{10} + 4590475306486 T^{11} + 45774690461867 T^{12} + 370255257704027 T^{13} + 3296694555307217 T^{14} + 24210188357158951 T^{15} + 3296694555307217 p T^{16} + 370255257704027 p^{2} T^{17} + 45774690461867 p^{3} T^{18} + 4590475306486 p^{4} T^{19} + 516847762290 p^{5} T^{20} + 45505261918 p^{6} T^{21} + 4668367919 p^{7} T^{22} + 350964306 p^{8} T^{23} + 32745051 p^{9} T^{24} + 2002547 p^{10} T^{25} + 169120 p^{11} T^{26} + 7639 p^{12} T^{27} + 579 p^{13} T^{28} + 15 p^{14} T^{29} + p^{15} T^{30} \) | |
| 61 | \( 1 + 25 T + 704 T^{2} + 10792 T^{3} + 177523 T^{4} + 1943180 T^{5} + 23525325 T^{6} + 193430908 T^{7} + 1952516273 T^{8} + 12622997338 T^{9} + 127543389090 T^{10} + 757669872721 T^{11} + 9026756735533 T^{12} + 56811990153815 T^{13} + 674740025556771 T^{14} + 3994392744853909 T^{15} + 674740025556771 p T^{16} + 56811990153815 p^{2} T^{17} + 9026756735533 p^{3} T^{18} + 757669872721 p^{4} T^{19} + 127543389090 p^{5} T^{20} + 12622997338 p^{6} T^{21} + 1952516273 p^{7} T^{22} + 193430908 p^{8} T^{23} + 23525325 p^{9} T^{24} + 1943180 p^{10} T^{25} + 177523 p^{11} T^{26} + 10792 p^{12} T^{27} + 704 p^{13} T^{28} + 25 p^{14} T^{29} + p^{15} T^{30} \) | |
| 67 | \( 1 + 13 T + 615 T^{2} + 7573 T^{3} + 191216 T^{4} + 2189352 T^{5} + 39536506 T^{6} + 417420925 T^{7} + 6054334790 T^{8} + 58787642605 T^{9} + 726487773199 T^{10} + 6486730964914 T^{11} + 70625392285309 T^{12} + 579608515096557 T^{13} + 5673193686048466 T^{14} + 42668316154857166 T^{15} + 5673193686048466 p T^{16} + 579608515096557 p^{2} T^{17} + 70625392285309 p^{3} T^{18} + 6486730964914 p^{4} T^{19} + 726487773199 p^{5} T^{20} + 58787642605 p^{6} T^{21} + 6054334790 p^{7} T^{22} + 417420925 p^{8} T^{23} + 39536506 p^{9} T^{24} + 2189352 p^{10} T^{25} + 191216 p^{11} T^{26} + 7573 p^{12} T^{27} + 615 p^{13} T^{28} + 13 p^{14} T^{29} + p^{15} T^{30} \) | |
| 71 | \( 1 + 4 T + 632 T^{2} + 502 T^{3} + 185458 T^{4} - 397282 T^{5} + 35541299 T^{6} - 163271909 T^{7} + 5235535797 T^{8} - 32070603741 T^{9} + 638736178666 T^{10} - 4230224382817 T^{11} + 65483336524014 T^{12} - 421528679761701 T^{13} + 5601772710058578 T^{14} - 33356118809364163 T^{15} + 5601772710058578 p T^{16} - 421528679761701 p^{2} T^{17} + 65483336524014 p^{3} T^{18} - 4230224382817 p^{4} T^{19} + 638736178666 p^{5} T^{20} - 32070603741 p^{6} T^{21} + 5235535797 p^{7} T^{22} - 163271909 p^{8} T^{23} + 35541299 p^{9} T^{24} - 397282 p^{10} T^{25} + 185458 p^{11} T^{26} + 502 p^{12} T^{27} + 632 p^{13} T^{28} + 4 p^{14} T^{29} + p^{15} T^{30} \) | |
| 73 | \( 1 + 4 T + 534 T^{2} + 264 T^{3} + 137538 T^{4} - 293362 T^{5} + 25166967 T^{6} - 91346815 T^{7} + 3706404247 T^{8} - 16091886435 T^{9} + 446323158314 T^{10} - 2103105094257 T^{11} + 44910222074378 T^{12} - 212567674424307 T^{13} + 3855276940661662 T^{14} - 17159692249320723 T^{15} + 3855276940661662 p T^{16} - 212567674424307 p^{2} T^{17} + 44910222074378 p^{3} T^{18} - 2103105094257 p^{4} T^{19} + 446323158314 p^{5} T^{20} - 16091886435 p^{6} T^{21} + 3706404247 p^{7} T^{22} - 91346815 p^{8} T^{23} + 25166967 p^{9} T^{24} - 293362 p^{10} T^{25} + 137538 p^{11} T^{26} + 264 p^{12} T^{27} + 534 p^{13} T^{28} + 4 p^{14} T^{29} + p^{15} T^{30} \) | |
| 79 | \( 1 + 20 T + 645 T^{2} + 9206 T^{3} + 183056 T^{4} + 2183110 T^{5} + 34889085 T^{6} + 371522371 T^{7} + 5154300356 T^{8} + 49838533168 T^{9} + 621804721213 T^{10} + 5543402686927 T^{11} + 63831606079410 T^{12} + 531330162731671 T^{13} + 5717700011767126 T^{14} + 44682110066151706 T^{15} + 5717700011767126 p T^{16} + 531330162731671 p^{2} T^{17} + 63831606079410 p^{3} T^{18} + 5543402686927 p^{4} T^{19} + 621804721213 p^{5} T^{20} + 49838533168 p^{6} T^{21} + 5154300356 p^{7} T^{22} + 371522371 p^{8} T^{23} + 34889085 p^{9} T^{24} + 2183110 p^{10} T^{25} + 183056 p^{11} T^{26} + 9206 p^{12} T^{27} + 645 p^{13} T^{28} + 20 p^{14} T^{29} + p^{15} T^{30} \) | |
| 83 | \( 1 - T + 438 T^{2} - 1184 T^{3} + 102828 T^{4} - 409267 T^{5} + 17735635 T^{6} - 88955919 T^{7} + 2458478535 T^{8} - 14750859586 T^{9} + 287970408162 T^{10} - 1947266391839 T^{11} + 29580981506760 T^{12} - 211069730754697 T^{13} + 2712441749211901 T^{14} - 19137772650184398 T^{15} + 2712441749211901 p T^{16} - 211069730754697 p^{2} T^{17} + 29580981506760 p^{3} T^{18} - 1947266391839 p^{4} T^{19} + 287970408162 p^{5} T^{20} - 14750859586 p^{6} T^{21} + 2458478535 p^{7} T^{22} - 88955919 p^{8} T^{23} + 17735635 p^{9} T^{24} - 409267 p^{10} T^{25} + 102828 p^{11} T^{26} - 1184 p^{12} T^{27} + 438 p^{13} T^{28} - p^{14} T^{29} + p^{15} T^{30} \) | |
| 89 | \( 1 + 41 T + 1383 T^{2} + 31516 T^{3} + 640048 T^{4} + 10630469 T^{5} + 164059755 T^{6} + 2213298705 T^{7} + 28427802424 T^{8} + 329112950683 T^{9} + 3694078482133 T^{10} + 38127453917062 T^{11} + 389915728956404 T^{12} + 3741872180833692 T^{13} + 36476607845872138 T^{14} + 339335712605184808 T^{15} + 36476607845872138 p T^{16} + 3741872180833692 p^{2} T^{17} + 389915728956404 p^{3} T^{18} + 38127453917062 p^{4} T^{19} + 3694078482133 p^{5} T^{20} + 329112950683 p^{6} T^{21} + 28427802424 p^{7} T^{22} + 2213298705 p^{8} T^{23} + 164059755 p^{9} T^{24} + 10630469 p^{10} T^{25} + 640048 p^{11} T^{26} + 31516 p^{12} T^{27} + 1383 p^{13} T^{28} + 41 p^{14} T^{29} + p^{15} T^{30} \) | |
| 97 | \( 1 + 57 T + 2003 T^{2} + 50546 T^{3} + 1038755 T^{4} + 18182899 T^{5} + 285735514 T^{6} + 4126872660 T^{7} + 56197885287 T^{8} + 725108304786 T^{9} + 8936897106837 T^{10} + 105001306864355 T^{11} + 1182904315916814 T^{12} + 12772656019517817 T^{13} + 132951014560221890 T^{14} + 1332215065696872301 T^{15} + 132951014560221890 p T^{16} + 12772656019517817 p^{2} T^{17} + 1182904315916814 p^{3} T^{18} + 105001306864355 p^{4} T^{19} + 8936897106837 p^{5} T^{20} + 725108304786 p^{6} T^{21} + 56197885287 p^{7} T^{22} + 4126872660 p^{8} T^{23} + 285735514 p^{9} T^{24} + 18182899 p^{10} T^{25} + 1038755 p^{11} T^{26} + 50546 p^{12} T^{27} + 2003 p^{13} T^{28} + 57 p^{14} T^{29} + p^{15} T^{30} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{30} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.32784074703082020303144573106, −2.32729972134856006642605988079, −2.32032311796766204677698902799, −2.30949902641556043652572826069, −2.30263555286705572656061944695, −2.26256997950853029206094934864, −2.19873375169484878708276394172, −2.02250583449842028343560963001, −2.02145526524805759381442342324, −1.92065829926548542985871499894, −1.65948048066210339528684189350, −1.65240045406891982754736385491, −1.62612584478655250246213036775, −1.53180774462012345684831564331, −1.50533095946239576829056062405, −1.41905324942858906604982612760, −1.40113026715751502222279800756, −1.36832165878406490913664355621, −1.24671740740159360699391844819, −1.19069984244191209177027219927, −1.13325173468721351225937605370, −1.13104078781894477721266189341, −0.977580836695719589652754714221, −0.939758677473102902484201776499, −0.78718281182167496016790352000, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.78718281182167496016790352000, 0.939758677473102902484201776499, 0.977580836695719589652754714221, 1.13104078781894477721266189341, 1.13325173468721351225937605370, 1.19069984244191209177027219927, 1.24671740740159360699391844819, 1.36832165878406490913664355621, 1.40113026715751502222279800756, 1.41905324942858906604982612760, 1.50533095946239576829056062405, 1.53180774462012345684831564331, 1.62612584478655250246213036775, 1.65240045406891982754736385491, 1.65948048066210339528684189350, 1.92065829926548542985871499894, 2.02145526524805759381442342324, 2.02250583449842028343560963001, 2.19873375169484878708276394172, 2.26256997950853029206094934864, 2.30263555286705572656061944695, 2.30949902641556043652572826069, 2.32032311796766204677698902799, 2.32729972134856006642605988079, 2.32784074703082020303144573106
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.