Dirichlet series
| L(s) = 1 | − 3-s + 5-s − 11·7-s − 14·9-s − 12·11-s − 10·13-s − 15-s + 15·17-s + 11·21-s − 21·23-s − 35·25-s + 3·27-s + 23·29-s − 31·31-s + 12·33-s − 11·35-s − 10·37-s + 10·39-s − 15·41-s − 3·43-s − 14·45-s − 47·47-s + 17·49-s − 15·51-s − 7·53-s − 12·55-s − 15·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 4.15·7-s − 4.66·9-s − 3.61·11-s − 2.77·13-s − 0.258·15-s + 3.63·17-s + 2.40·21-s − 4.37·23-s − 7·25-s + 0.577·27-s + 4.27·29-s − 5.56·31-s + 2.08·33-s − 1.85·35-s − 1.64·37-s + 1.60·39-s − 2.34·41-s − 0.457·43-s − 2.08·45-s − 6.85·47-s + 17/7·49-s − 2.10·51-s − 0.961·53-s − 1.61·55-s − 1.95·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 17^{15} \cdot 59^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 17^{15} \cdot 59^{15}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(30\) |
| Conductor: | \(2^{30} \cdot 17^{15} \cdot 59^{15}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.84211\times 10^{22}\) |
| Root analytic conductor: | \(5.66003\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(15\) |
| Selberg data: | \((30,\ 2^{30} \cdot 17^{15} \cdot 59^{15} ,\ ( \ : [1/2]^{15} ),\ -1 )\) |
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 | \( 1 \) |
| 17 | \( ( 1 - T )^{15} \) | |
| 59 | \( ( 1 + T )^{15} \) | |
| good | 3 | \( 1 + T + 5 p T^{2} + 26 T^{3} + 47 p T^{4} + 94 p T^{5} + 113 p^{2} T^{6} + 2096 T^{7} + 8 p^{6} T^{8} + 12052 T^{9} + 27823 T^{10} + 55768 T^{11} + 113005 T^{12} + 215053 T^{13} + 14536 p^{3} T^{14} + 233884 p T^{15} + 14536 p^{4} T^{16} + 215053 p^{2} T^{17} + 113005 p^{3} T^{18} + 55768 p^{4} T^{19} + 27823 p^{5} T^{20} + 12052 p^{6} T^{21} + 8 p^{13} T^{22} + 2096 p^{8} T^{23} + 113 p^{11} T^{24} + 94 p^{11} T^{25} + 47 p^{12} T^{26} + 26 p^{12} T^{27} + 5 p^{14} T^{28} + p^{14} T^{29} + p^{15} T^{30} \) |
| 5 | \( 1 - T + 36 T^{2} - 34 T^{3} + 652 T^{4} - 602 T^{5} + 7984 T^{6} - 1493 p T^{7} + 75179 T^{8} - 14346 p T^{9} + 582334 T^{10} - 556052 T^{11} + 3842951 T^{12} - 3564841 T^{13} + 21988347 T^{14} - 19305638 T^{15} + 21988347 p T^{16} - 3564841 p^{2} T^{17} + 3842951 p^{3} T^{18} - 556052 p^{4} T^{19} + 582334 p^{5} T^{20} - 14346 p^{7} T^{21} + 75179 p^{7} T^{22} - 1493 p^{9} T^{23} + 7984 p^{9} T^{24} - 602 p^{10} T^{25} + 652 p^{11} T^{26} - 34 p^{12} T^{27} + 36 p^{13} T^{28} - p^{14} T^{29} + p^{15} T^{30} \) | |
| 7 | \( 1 + 11 T + 104 T^{2} + 688 T^{3} + 12 p^{3} T^{4} + 20763 T^{5} + 97477 T^{6} + 410898 T^{7} + 1637282 T^{8} + 860794 p T^{9} + 21140109 T^{10} + 69476752 T^{11} + 31239998 p T^{12} + 649484690 T^{13} + 1851901833 T^{14} + 4998625284 T^{15} + 1851901833 p T^{16} + 649484690 p^{2} T^{17} + 31239998 p^{4} T^{18} + 69476752 p^{4} T^{19} + 21140109 p^{5} T^{20} + 860794 p^{7} T^{21} + 1637282 p^{7} T^{22} + 410898 p^{8} T^{23} + 97477 p^{9} T^{24} + 20763 p^{10} T^{25} + 12 p^{14} T^{26} + 688 p^{12} T^{27} + 104 p^{13} T^{28} + 11 p^{14} T^{29} + p^{15} T^{30} \) | |
| 11 | \( 1 + 12 T + 133 T^{2} + 1047 T^{3} + 7530 T^{4} + 46565 T^{5} + 266674 T^{6} + 1388283 T^{7} + 6785876 T^{8} + 30896141 T^{9} + 12130231 p T^{10} + 543813670 T^{11} + 2116387758 T^{12} + 7831681555 T^{13} + 27783012053 T^{14} + 94052272884 T^{15} + 27783012053 p T^{16} + 7831681555 p^{2} T^{17} + 2116387758 p^{3} T^{18} + 543813670 p^{4} T^{19} + 12130231 p^{6} T^{20} + 30896141 p^{6} T^{21} + 6785876 p^{7} T^{22} + 1388283 p^{8} T^{23} + 266674 p^{9} T^{24} + 46565 p^{10} T^{25} + 7530 p^{11} T^{26} + 1047 p^{12} T^{27} + 133 p^{13} T^{28} + 12 p^{14} T^{29} + p^{15} T^{30} \) | |
| 13 | \( 1 + 10 T + 140 T^{2} + 994 T^{3} + 8547 T^{4} + 48921 T^{5} + 326579 T^{6} + 1601994 T^{7} + 9073914 T^{8} + 39552951 T^{9} + 198844678 T^{10} + 60568136 p T^{11} + 3597109908 T^{12} + 13089054362 T^{13} + 54935615381 T^{14} + 184451327824 T^{15} + 54935615381 p T^{16} + 13089054362 p^{2} T^{17} + 3597109908 p^{3} T^{18} + 60568136 p^{5} T^{19} + 198844678 p^{5} T^{20} + 39552951 p^{6} T^{21} + 9073914 p^{7} T^{22} + 1601994 p^{8} T^{23} + 326579 p^{9} T^{24} + 48921 p^{10} T^{25} + 8547 p^{11} T^{26} + 994 p^{12} T^{27} + 140 p^{13} T^{28} + 10 p^{14} T^{29} + p^{15} T^{30} \) | |
| 19 | \( 1 + 83 T^{2} + 97 T^{3} + 4154 T^{4} + 9003 T^{5} + 152790 T^{6} + 455226 T^{7} + 4711130 T^{8} + 16151252 T^{9} + 126895345 T^{10} + 448984987 T^{11} + 3060918579 T^{12} + 10411079176 T^{13} + 65732842410 T^{14} + 210170897854 T^{15} + 65732842410 p T^{16} + 10411079176 p^{2} T^{17} + 3060918579 p^{3} T^{18} + 448984987 p^{4} T^{19} + 126895345 p^{5} T^{20} + 16151252 p^{6} T^{21} + 4711130 p^{7} T^{22} + 455226 p^{8} T^{23} + 152790 p^{9} T^{24} + 9003 p^{10} T^{25} + 4154 p^{11} T^{26} + 97 p^{12} T^{27} + 83 p^{13} T^{28} + p^{15} T^{30} \) | |
| 23 | \( 1 + 21 T + 347 T^{2} + 4336 T^{3} + 46198 T^{4} + 18890 p T^{5} + 3675866 T^{6} + 28621248 T^{7} + 206719642 T^{8} + 1397159765 T^{9} + 8896862239 T^{10} + 53499099763 T^{11} + 305250702792 T^{12} + 1654375544921 T^{13} + 8532394903305 T^{14} + 1823085499650 p T^{15} + 8532394903305 p T^{16} + 1654375544921 p^{2} T^{17} + 305250702792 p^{3} T^{18} + 53499099763 p^{4} T^{19} + 8896862239 p^{5} T^{20} + 1397159765 p^{6} T^{21} + 206719642 p^{7} T^{22} + 28621248 p^{8} T^{23} + 3675866 p^{9} T^{24} + 18890 p^{11} T^{25} + 46198 p^{11} T^{26} + 4336 p^{12} T^{27} + 347 p^{13} T^{28} + 21 p^{14} T^{29} + p^{15} T^{30} \) | |
| 29 | \( 1 - 23 T + 427 T^{2} - 5461 T^{3} + 61354 T^{4} - 569838 T^{5} + 4861036 T^{6} - 36739478 T^{7} + 263280045 T^{8} - 1739117478 T^{9} + 11146165520 T^{10} - 67433048860 T^{11} + 401224859086 T^{12} - 78673107003 p T^{13} + 12841014957963 T^{14} - 69439016441342 T^{15} + 12841014957963 p T^{16} - 78673107003 p^{3} T^{17} + 401224859086 p^{3} T^{18} - 67433048860 p^{4} T^{19} + 11146165520 p^{5} T^{20} - 1739117478 p^{6} T^{21} + 263280045 p^{7} T^{22} - 36739478 p^{8} T^{23} + 4861036 p^{9} T^{24} - 569838 p^{10} T^{25} + 61354 p^{11} T^{26} - 5461 p^{12} T^{27} + 427 p^{13} T^{28} - 23 p^{14} T^{29} + p^{15} T^{30} \) | |
| 31 | \( 1 + p T + 692 T^{2} + 11612 T^{3} + 165720 T^{4} + 2039159 T^{5} + 22496680 T^{6} + 224081837 T^{7} + 2050852122 T^{8} + 17313998075 T^{9} + 136131839684 T^{10} + 998684238486 T^{11} + 6875238587605 T^{12} + 44437839105999 T^{13} + 270594191528358 T^{14} + 1551242144539090 T^{15} + 270594191528358 p T^{16} + 44437839105999 p^{2} T^{17} + 6875238587605 p^{3} T^{18} + 998684238486 p^{4} T^{19} + 136131839684 p^{5} T^{20} + 17313998075 p^{6} T^{21} + 2050852122 p^{7} T^{22} + 224081837 p^{8} T^{23} + 22496680 p^{9} T^{24} + 2039159 p^{10} T^{25} + 165720 p^{11} T^{26} + 11612 p^{12} T^{27} + 692 p^{13} T^{28} + p^{15} T^{29} + p^{15} T^{30} \) | |
| 37 | \( 1 + 10 T + 312 T^{2} + 84 p T^{3} + 52581 T^{4} + 483244 T^{5} + 6034705 T^{6} + 50418088 T^{7} + 516444781 T^{8} + 3928176048 T^{9} + 34696477720 T^{10} + 240745761242 T^{11} + 1885028766162 T^{12} + 11943747411556 T^{13} + 84189154828618 T^{14} + 486619808224408 T^{15} + 84189154828618 p T^{16} + 11943747411556 p^{2} T^{17} + 1885028766162 p^{3} T^{18} + 240745761242 p^{4} T^{19} + 34696477720 p^{5} T^{20} + 3928176048 p^{6} T^{21} + 516444781 p^{7} T^{22} + 50418088 p^{8} T^{23} + 6034705 p^{9} T^{24} + 483244 p^{10} T^{25} + 52581 p^{11} T^{26} + 84 p^{13} T^{27} + 312 p^{13} T^{28} + 10 p^{14} T^{29} + p^{15} T^{30} \) | |
| 41 | \( 1 + 15 T + 442 T^{2} + 5223 T^{3} + 90894 T^{4} + 911629 T^{5} + 12025392 T^{6} + 106214266 T^{7} + 1162087852 T^{8} + 9210320951 T^{9} + 87305614458 T^{10} + 627124168079 T^{11} + 5277791151653 T^{12} + 34520123652249 T^{13} + 261648125984996 T^{14} + 1559014176372088 T^{15} + 261648125984996 p T^{16} + 34520123652249 p^{2} T^{17} + 5277791151653 p^{3} T^{18} + 627124168079 p^{4} T^{19} + 87305614458 p^{5} T^{20} + 9210320951 p^{6} T^{21} + 1162087852 p^{7} T^{22} + 106214266 p^{8} T^{23} + 12025392 p^{9} T^{24} + 911629 p^{10} T^{25} + 90894 p^{11} T^{26} + 5223 p^{12} T^{27} + 442 p^{13} T^{28} + 15 p^{14} T^{29} + p^{15} T^{30} \) | |
| 43 | \( 1 + 3 T + 278 T^{2} + 514 T^{3} + 38864 T^{4} + 25385 T^{5} + 3718122 T^{6} - 1286068 T^{7} + 282546143 T^{8} - 278051405 T^{9} + 18321423239 T^{10} - 23233496517 T^{11} + 1033949685168 T^{12} - 1384563262343 T^{13} + 50876286219341 T^{14} - 65935465846962 T^{15} + 50876286219341 p T^{16} - 1384563262343 p^{2} T^{17} + 1033949685168 p^{3} T^{18} - 23233496517 p^{4} T^{19} + 18321423239 p^{5} T^{20} - 278051405 p^{6} T^{21} + 282546143 p^{7} T^{22} - 1286068 p^{8} T^{23} + 3718122 p^{9} T^{24} + 25385 p^{10} T^{25} + 38864 p^{11} T^{26} + 514 p^{12} T^{27} + 278 p^{13} T^{28} + 3 p^{14} T^{29} + p^{15} T^{30} \) | |
| 47 | \( 1 + p T + 1360 T^{2} + 29242 T^{3} + 515710 T^{4} + 7806435 T^{5} + 104570527 T^{6} + 1263761173 T^{7} + 13972610942 T^{8} + 142749487480 T^{9} + 1357652693882 T^{10} + 12087146093867 T^{11} + 101151189464855 T^{12} + 798016923155622 T^{13} + 5947378574242063 T^{14} + 41918381294391180 T^{15} + 5947378574242063 p T^{16} + 798016923155622 p^{2} T^{17} + 101151189464855 p^{3} T^{18} + 12087146093867 p^{4} T^{19} + 1357652693882 p^{5} T^{20} + 142749487480 p^{6} T^{21} + 13972610942 p^{7} T^{22} + 1263761173 p^{8} T^{23} + 104570527 p^{9} T^{24} + 7806435 p^{10} T^{25} + 515710 p^{11} T^{26} + 29242 p^{12} T^{27} + 1360 p^{13} T^{28} + p^{15} T^{29} + p^{15} T^{30} \) | |
| 53 | \( 1 + 7 T + 196 T^{2} + 608 T^{3} + 21087 T^{4} + 23332 T^{5} + 1863719 T^{6} - 1169126 T^{7} + 2777031 p T^{8} - 208121852 T^{9} + 10894186148 T^{10} - 18900938535 T^{11} + 715640854992 T^{12} - 25634689282 p T^{13} + 41942713891806 T^{14} - 80813786672112 T^{15} + 41942713891806 p T^{16} - 25634689282 p^{3} T^{17} + 715640854992 p^{3} T^{18} - 18900938535 p^{4} T^{19} + 10894186148 p^{5} T^{20} - 208121852 p^{6} T^{21} + 2777031 p^{8} T^{22} - 1169126 p^{8} T^{23} + 1863719 p^{9} T^{24} + 23332 p^{10} T^{25} + 21087 p^{11} T^{26} + 608 p^{12} T^{27} + 196 p^{13} T^{28} + 7 p^{14} T^{29} + p^{15} T^{30} \) | |
| 61 | \( 1 - T + 535 T^{2} + 896 T^{3} + 129742 T^{4} + 619919 T^{5} + 19695042 T^{6} + 159104993 T^{7} + 2271075096 T^{8} + 23702591530 T^{9} + 229839047737 T^{10} + 2401962078682 T^{11} + 21008186472825 T^{12} + 185309068314783 T^{13} + 1616722061410274 T^{14} + 12035047966225248 T^{15} + 1616722061410274 p T^{16} + 185309068314783 p^{2} T^{17} + 21008186472825 p^{3} T^{18} + 2401962078682 p^{4} T^{19} + 229839047737 p^{5} T^{20} + 23702591530 p^{6} T^{21} + 2271075096 p^{7} T^{22} + 159104993 p^{8} T^{23} + 19695042 p^{9} T^{24} + 619919 p^{10} T^{25} + 129742 p^{11} T^{26} + 896 p^{12} T^{27} + 535 p^{13} T^{28} - p^{14} T^{29} + p^{15} T^{30} \) | |
| 67 | \( 1 + 20 T + 715 T^{2} + 10412 T^{3} + 216322 T^{4} + 2479018 T^{5} + 38923425 T^{6} + 370303811 T^{7} + 4952459444 T^{8} + 41220861511 T^{9} + 506297442528 T^{10} + 3870640116092 T^{11} + 44987121763246 T^{12} + 322484235897585 T^{13} + 3508819206645825 T^{14} + 23386315081882382 T^{15} + 3508819206645825 p T^{16} + 322484235897585 p^{2} T^{17} + 44987121763246 p^{3} T^{18} + 3870640116092 p^{4} T^{19} + 506297442528 p^{5} T^{20} + 41220861511 p^{6} T^{21} + 4952459444 p^{7} T^{22} + 370303811 p^{8} T^{23} + 38923425 p^{9} T^{24} + 2479018 p^{10} T^{25} + 216322 p^{11} T^{26} + 10412 p^{12} T^{27} + 715 p^{13} T^{28} + 20 p^{14} T^{29} + p^{15} T^{30} \) | |
| 71 | \( 1 + 13 T + 734 T^{2} + 8444 T^{3} + 256616 T^{4} + 2655901 T^{5} + 57716900 T^{6} + 543971994 T^{7} + 9469919098 T^{8} + 81876459729 T^{9} + 1210704385834 T^{10} + 9631650796678 T^{11} + 125115095370877 T^{12} + 915416527813805 T^{13} + 10666306815567054 T^{14} + 71502153804051136 T^{15} + 10666306815567054 p T^{16} + 915416527813805 p^{2} T^{17} + 125115095370877 p^{3} T^{18} + 9631650796678 p^{4} T^{19} + 1210704385834 p^{5} T^{20} + 81876459729 p^{6} T^{21} + 9469919098 p^{7} T^{22} + 543971994 p^{8} T^{23} + 57716900 p^{9} T^{24} + 2655901 p^{10} T^{25} + 256616 p^{11} T^{26} + 8444 p^{12} T^{27} + 734 p^{13} T^{28} + 13 p^{14} T^{29} + p^{15} T^{30} \) | |
| 73 | \( 1 + 24 T + 819 T^{2} + 15192 T^{3} + 309951 T^{4} + 4704095 T^{5} + 72983727 T^{6} + 942396132 T^{7} + 12143149592 T^{8} + 137237813352 T^{9} + 1539029575583 T^{10} + 15575909116591 T^{11} + 156785812756212 T^{12} + 1448205613996866 T^{13} + 13364742955882919 T^{14} + 114116433932227924 T^{15} + 13364742955882919 p T^{16} + 1448205613996866 p^{2} T^{17} + 156785812756212 p^{3} T^{18} + 15575909116591 p^{4} T^{19} + 1539029575583 p^{5} T^{20} + 137237813352 p^{6} T^{21} + 12143149592 p^{7} T^{22} + 942396132 p^{8} T^{23} + 72983727 p^{9} T^{24} + 4704095 p^{10} T^{25} + 309951 p^{11} T^{26} + 15192 p^{12} T^{27} + 819 p^{13} T^{28} + 24 p^{14} T^{29} + p^{15} T^{30} \) | |
| 79 | \( 1 + 34 T + 1173 T^{2} + 26305 T^{3} + 562443 T^{4} + 9836766 T^{5} + 162873092 T^{6} + 2382471793 T^{7} + 33050635472 T^{8} + 419254986829 T^{9} + 5061673727393 T^{10} + 56862498191110 T^{11} + 609952841001422 T^{12} + 77754884712001 p T^{13} + 59197444488140998 T^{14} + 537655417992347836 T^{15} + 59197444488140998 p T^{16} + 77754884712001 p^{3} T^{17} + 609952841001422 p^{3} T^{18} + 56862498191110 p^{4} T^{19} + 5061673727393 p^{5} T^{20} + 419254986829 p^{6} T^{21} + 33050635472 p^{7} T^{22} + 2382471793 p^{8} T^{23} + 162873092 p^{9} T^{24} + 9836766 p^{10} T^{25} + 562443 p^{11} T^{26} + 26305 p^{12} T^{27} + 1173 p^{13} T^{28} + 34 p^{14} T^{29} + p^{15} T^{30} \) | |
| 83 | \( 1 + 40 T + 1031 T^{2} + 17725 T^{3} + 256332 T^{4} + 3200681 T^{5} + 40303961 T^{6} + 474910242 T^{7} + 5415040583 T^{8} + 55001306684 T^{9} + 557287568498 T^{10} + 5451190530076 T^{11} + 55382564648974 T^{12} + 518249610734501 T^{13} + 4822234874235804 T^{14} + 42594893289671942 T^{15} + 4822234874235804 p T^{16} + 518249610734501 p^{2} T^{17} + 55382564648974 p^{3} T^{18} + 5451190530076 p^{4} T^{19} + 557287568498 p^{5} T^{20} + 55001306684 p^{6} T^{21} + 5415040583 p^{7} T^{22} + 474910242 p^{8} T^{23} + 40303961 p^{9} T^{24} + 3200681 p^{10} T^{25} + 256332 p^{11} T^{26} + 17725 p^{12} T^{27} + 1031 p^{13} T^{28} + 40 p^{14} T^{29} + p^{15} T^{30} \) | |
| 89 | \( 1 + 46 T + 1724 T^{2} + 45608 T^{3} + 1055696 T^{4} + 20603015 T^{5} + 365472591 T^{6} + 5803698716 T^{7} + 85671363923 T^{8} + 1164978384166 T^{9} + 14926872643907 T^{10} + 178875659973247 T^{11} + 2035217727608960 T^{12} + 21823550842118395 T^{13} + 222987948881550762 T^{14} + 2153948747947025494 T^{15} + 222987948881550762 p T^{16} + 21823550842118395 p^{2} T^{17} + 2035217727608960 p^{3} T^{18} + 178875659973247 p^{4} T^{19} + 14926872643907 p^{5} T^{20} + 1164978384166 p^{6} T^{21} + 85671363923 p^{7} T^{22} + 5803698716 p^{8} T^{23} + 365472591 p^{9} T^{24} + 20603015 p^{10} T^{25} + 1055696 p^{11} T^{26} + 45608 p^{12} T^{27} + 1724 p^{13} T^{28} + 46 p^{14} T^{29} + p^{15} T^{30} \) | |
| 97 | \( 1 + 5 T + 860 T^{2} + 3654 T^{3} + 357585 T^{4} + 1321128 T^{5} + 96179967 T^{6} + 315767895 T^{7} + 18923279992 T^{8} + 56408221506 T^{9} + 2929154946806 T^{10} + 8083074985785 T^{11} + 375649080319275 T^{12} + 972628985193628 T^{13} + 41452416754446442 T^{14} + 101048739140589122 T^{15} + 41452416754446442 p T^{16} + 972628985193628 p^{2} T^{17} + 375649080319275 p^{3} T^{18} + 8083074985785 p^{4} T^{19} + 2929154946806 p^{5} T^{20} + 56408221506 p^{6} T^{21} + 18923279992 p^{7} T^{22} + 315767895 p^{8} T^{23} + 96179967 p^{9} T^{24} + 1321128 p^{10} T^{25} + 357585 p^{11} T^{26} + 3654 p^{12} T^{27} + 860 p^{13} T^{28} + 5 p^{14} T^{29} + p^{15} T^{30} \) | |
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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
−2.60464482561605444619438728396, −2.59286244573114342292827545799, −2.59149363152728147917498971864, −2.50612260616669932256816005027, −2.49842014425133489811508590603, −2.45128344908059588271171862670, −2.28914489985431473630714034302, −2.24864574689410244526664039045, −2.24150901818968277694386536918, −2.00945012594455727412844236683, −1.92825664505044251453012629766, −1.78964492882328685774297702936, −1.78001730447424261741527438755, −1.72961397883709766709091805814, −1.64834261361765959550628914810, −1.62220762582975305310228970752, −1.48751109445951892465805851459, −1.46013286503278680144329230978, −1.33644365673593722320694075076, −1.32199445430100795155136140779, −1.30252368267458884295331520572, −1.24790195271415593322700143825, −1.18695606210322690576879922610, −1.15780524866619523343240294933, −0.75710249449547025861077381000, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.75710249449547025861077381000, 1.15780524866619523343240294933, 1.18695606210322690576879922610, 1.24790195271415593322700143825, 1.30252368267458884295331520572, 1.32199445430100795155136140779, 1.33644365673593722320694075076, 1.46013286503278680144329230978, 1.48751109445951892465805851459, 1.62220762582975305310228970752, 1.64834261361765959550628914810, 1.72961397883709766709091805814, 1.78001730447424261741527438755, 1.78964492882328685774297702936, 1.92825664505044251453012629766, 2.00945012594455727412844236683, 2.24150901818968277694386536918, 2.24864574689410244526664039045, 2.28914489985431473630714034302, 2.45128344908059588271171862670, 2.49842014425133489811508590603, 2.50612260616669932256816005027, 2.59149363152728147917498971864, 2.59286244573114342292827545799, 2.60464482561605444619438728396
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.