Dirichlet series
| L(s) = 1 | + 15·2-s + 2·3-s + 120·4-s − 15·5-s + 30·6-s + 15·7-s + 680·8-s − 8·9-s − 225·10-s + 13·11-s + 240·12-s + 3·13-s + 225·14-s − 30·15-s + 3.06e3·16-s + 11·17-s − 120·18-s + 4·19-s − 1.80e3·20-s + 30·21-s + 195·22-s + 21·23-s + 1.36e3·24-s + 120·25-s + 45·26-s − 18·27-s + 1.80e3·28-s + ⋯ |
| L(s) = 1 | + 10.6·2-s + 1.15·3-s + 60·4-s − 6.70·5-s + 12.2·6-s + 5.66·7-s + 240.·8-s − 8/3·9-s − 71.1·10-s + 3.91·11-s + 69.2·12-s + 0.832·13-s + 60.1·14-s − 7.74·15-s + 765·16-s + 2.66·17-s − 28.2·18-s + 0.917·19-s − 402.·20-s + 6.54·21-s + 41.5·22-s + 4.37·23-s + 277.·24-s + 24·25-s + 8.82·26-s − 3.46·27-s + 340.·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{15} \cdot 7^{15} \cdot 97^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{15} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{15} \cdot 7^{15} \cdot 97^{15}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{15} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(30\) |
| Conductor: | \(2^{15} \cdot 5^{15} \cdot 7^{15} \cdot 97^{15}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.02849\times 10^{26}\) |
| Root analytic conductor: | \(7.36331\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((30,\ 2^{15} \cdot 5^{15} \cdot 7^{15} \cdot 97^{15} ,\ ( \ : [1/2]^{15} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.576088605\times10^{7}\) |
| \(L(\frac12)\) | \(\approx\) | \(1.576088605\times10^{7}\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{15} \) |
| 5 | \( ( 1 + T )^{15} \) | |
| 7 | \( ( 1 - T )^{15} \) | |
| 97 | \( ( 1 - T )^{15} \) | |
| good | 3 | \( 1 - 2 T + 4 p T^{2} - 22 T^{3} + 25 p T^{4} - 14 p^{2} T^{5} + 356 T^{6} - 572 T^{7} + 2 p^{6} T^{8} - 2374 T^{9} + 1799 p T^{10} - 8858 T^{11} + 18478 T^{12} - 30202 T^{13} + 19733 p T^{14} - 95176 T^{15} + 19733 p^{2} T^{16} - 30202 p^{2} T^{17} + 18478 p^{3} T^{18} - 8858 p^{4} T^{19} + 1799 p^{6} T^{20} - 2374 p^{6} T^{21} + 2 p^{13} T^{22} - 572 p^{8} T^{23} + 356 p^{9} T^{24} - 14 p^{12} T^{25} + 25 p^{12} T^{26} - 22 p^{12} T^{27} + 4 p^{14} T^{28} - 2 p^{14} T^{29} + p^{15} T^{30} \) |
| 11 | \( 1 - 13 T + 151 T^{2} - 1217 T^{3} + 8916 T^{4} - 55331 T^{5} + 319318 T^{6} - 151444 p T^{7} + 8199102 T^{8} - 37496369 T^{9} + 163143468 T^{10} - 668239959 T^{11} + 2617840695 T^{12} - 80318367 p^{2} T^{13} + 34612322585 T^{14} - 117145277064 T^{15} + 34612322585 p T^{16} - 80318367 p^{4} T^{17} + 2617840695 p^{3} T^{18} - 668239959 p^{4} T^{19} + 163143468 p^{5} T^{20} - 37496369 p^{6} T^{21} + 8199102 p^{7} T^{22} - 151444 p^{9} T^{23} + 319318 p^{9} T^{24} - 55331 p^{10} T^{25} + 8916 p^{11} T^{26} - 1217 p^{12} T^{27} + 151 p^{13} T^{28} - 13 p^{14} T^{29} + p^{15} T^{30} \) | |
| 13 | \( 1 - 3 T + 90 T^{2} - 218 T^{3} + 3959 T^{4} - 616 p T^{5} + 118861 T^{6} - 209896 T^{7} + 215156 p T^{8} - 4433651 T^{9} + 4211301 p T^{10} - 79181328 T^{11} + 921003854 T^{12} - 1238832878 T^{13} + 13583670366 T^{14} - 17156780916 T^{15} + 13583670366 p T^{16} - 1238832878 p^{2} T^{17} + 921003854 p^{3} T^{18} - 79181328 p^{4} T^{19} + 4211301 p^{6} T^{20} - 4433651 p^{6} T^{21} + 215156 p^{8} T^{22} - 209896 p^{8} T^{23} + 118861 p^{9} T^{24} - 616 p^{11} T^{25} + 3959 p^{11} T^{26} - 218 p^{12} T^{27} + 90 p^{13} T^{28} - 3 p^{14} T^{29} + p^{15} T^{30} \) | |
| 17 | \( 1 - 11 T + 183 T^{2} - 1573 T^{3} + 15924 T^{4} - 113823 T^{5} + 884172 T^{6} - 320744 p T^{7} + 35385012 T^{8} - 192868481 T^{9} + 1090982512 T^{10} - 5343370171 T^{11} + 27005113489 T^{12} - 120152315565 T^{13} + 550509163747 T^{14} - 2237251726624 T^{15} + 550509163747 p T^{16} - 120152315565 p^{2} T^{17} + 27005113489 p^{3} T^{18} - 5343370171 p^{4} T^{19} + 1090982512 p^{5} T^{20} - 192868481 p^{6} T^{21} + 35385012 p^{7} T^{22} - 320744 p^{9} T^{23} + 884172 p^{9} T^{24} - 113823 p^{10} T^{25} + 15924 p^{11} T^{26} - 1573 p^{12} T^{27} + 183 p^{13} T^{28} - 11 p^{14} T^{29} + p^{15} T^{30} \) | |
| 19 | \( 1 - 4 T + 142 T^{2} - 422 T^{3} + 9525 T^{4} - 20728 T^{5} + 423402 T^{6} - 697140 T^{7} + 14742952 T^{8} - 20116084 T^{9} + 432546243 T^{10} - 521586922 T^{11} + 10843776020 T^{12} - 11666913712 T^{13} + 234590805983 T^{14} - 231096279288 T^{15} + 234590805983 p T^{16} - 11666913712 p^{2} T^{17} + 10843776020 p^{3} T^{18} - 521586922 p^{4} T^{19} + 432546243 p^{5} T^{20} - 20116084 p^{6} T^{21} + 14742952 p^{7} T^{22} - 697140 p^{8} T^{23} + 423402 p^{9} T^{24} - 20728 p^{10} T^{25} + 9525 p^{11} T^{26} - 422 p^{12} T^{27} + 142 p^{13} T^{28} - 4 p^{14} T^{29} + p^{15} T^{30} \) | |
| 23 | \( 1 - 21 T + 416 T^{2} - 5148 T^{3} + 59355 T^{4} - 524300 T^{5} + 4382131 T^{6} - 29950754 T^{7} + 198533488 T^{8} - 1107526599 T^{9} + 6241625233 T^{10} - 30148643334 T^{11} + 156348731054 T^{12} - 703802799796 T^{13} + 3603977903542 T^{14} - 16033292963264 T^{15} + 3603977903542 p T^{16} - 703802799796 p^{2} T^{17} + 156348731054 p^{3} T^{18} - 30148643334 p^{4} T^{19} + 6241625233 p^{5} T^{20} - 1107526599 p^{6} T^{21} + 198533488 p^{7} T^{22} - 29950754 p^{8} T^{23} + 4382131 p^{9} T^{24} - 524300 p^{10} T^{25} + 59355 p^{11} T^{26} - 5148 p^{12} T^{27} + 416 p^{13} T^{28} - 21 p^{14} T^{29} + p^{15} T^{30} \) | |
| 29 | \( 1 - 18 T + 358 T^{2} - 4140 T^{3} + 50501 T^{4} - 455330 T^{5} + 4331763 T^{6} - 33198542 T^{7} + 269782578 T^{8} - 1837948274 T^{9} + 13304561005 T^{10} - 82329491102 T^{11} + 541215452218 T^{12} - 3071326941418 T^{13} + 18509683113248 T^{14} - 3335370542064 p T^{15} + 18509683113248 p T^{16} - 3071326941418 p^{2} T^{17} + 541215452218 p^{3} T^{18} - 82329491102 p^{4} T^{19} + 13304561005 p^{5} T^{20} - 1837948274 p^{6} T^{21} + 269782578 p^{7} T^{22} - 33198542 p^{8} T^{23} + 4331763 p^{9} T^{24} - 455330 p^{10} T^{25} + 50501 p^{11} T^{26} - 4140 p^{12} T^{27} + 358 p^{13} T^{28} - 18 p^{14} T^{29} + p^{15} T^{30} \) | |
| 31 | \( 1 - 4 T + 249 T^{2} - 1008 T^{3} + 29291 T^{4} - 116124 T^{5} + 2154999 T^{6} - 8013300 T^{7} + 110527259 T^{8} - 363393148 T^{9} + 4198056007 T^{10} - 11237374320 T^{11} + 126362768613 T^{12} - 253910647268 T^{13} + 3481577166517 T^{14} - 6074974649688 T^{15} + 3481577166517 p T^{16} - 253910647268 p^{2} T^{17} + 126362768613 p^{3} T^{18} - 11237374320 p^{4} T^{19} + 4198056007 p^{5} T^{20} - 363393148 p^{6} T^{21} + 110527259 p^{7} T^{22} - 8013300 p^{8} T^{23} + 2154999 p^{9} T^{24} - 116124 p^{10} T^{25} + 29291 p^{11} T^{26} - 1008 p^{12} T^{27} + 249 p^{13} T^{28} - 4 p^{14} T^{29} + p^{15} T^{30} \) | |
| 37 | \( 1 - 16 T + 422 T^{2} - 4400 T^{3} + 67395 T^{4} - 493758 T^{5} + 5780396 T^{6} - 29116316 T^{7} + 8486128 p T^{8} - 944545214 T^{9} + 12923917857 T^{10} - 16469528164 T^{11} + 516294203302 T^{12} - 230169536988 T^{13} + 21414735351547 T^{14} - 7415891053776 T^{15} + 21414735351547 p T^{16} - 230169536988 p^{2} T^{17} + 516294203302 p^{3} T^{18} - 16469528164 p^{4} T^{19} + 12923917857 p^{5} T^{20} - 944545214 p^{6} T^{21} + 8486128 p^{8} T^{22} - 29116316 p^{8} T^{23} + 5780396 p^{9} T^{24} - 493758 p^{10} T^{25} + 67395 p^{11} T^{26} - 4400 p^{12} T^{27} + 422 p^{13} T^{28} - 16 p^{14} T^{29} + p^{15} T^{30} \) | |
| 41 | \( 1 - 3 T + 294 T^{2} - 1486 T^{3} + 46051 T^{4} - 284554 T^{5} + 5166541 T^{6} - 33603766 T^{7} + 450606412 T^{8} - 2902648929 T^{9} + 31655739713 T^{10} - 196485397648 T^{11} + 1837332369744 T^{12} - 10756262474794 T^{13} + 89405057679692 T^{14} - 484594831325672 T^{15} + 89405057679692 p T^{16} - 10756262474794 p^{2} T^{17} + 1837332369744 p^{3} T^{18} - 196485397648 p^{4} T^{19} + 31655739713 p^{5} T^{20} - 2902648929 p^{6} T^{21} + 450606412 p^{7} T^{22} - 33603766 p^{8} T^{23} + 5166541 p^{9} T^{24} - 284554 p^{10} T^{25} + 46051 p^{11} T^{26} - 1486 p^{12} T^{27} + 294 p^{13} T^{28} - 3 p^{14} T^{29} + p^{15} T^{30} \) | |
| 43 | \( 1 - 17 T + 468 T^{2} - 5832 T^{3} + 96745 T^{4} - 995064 T^{5} + 12733121 T^{6} - 114365838 T^{7} + 1231572270 T^{8} - 9923126905 T^{9} + 93621757147 T^{10} - 685779237772 T^{11} + 5795067696618 T^{12} - 38835052142582 T^{13} + 297723838725210 T^{14} - 1827809845879084 T^{15} + 297723838725210 p T^{16} - 38835052142582 p^{2} T^{17} + 5795067696618 p^{3} T^{18} - 685779237772 p^{4} T^{19} + 93621757147 p^{5} T^{20} - 9923126905 p^{6} T^{21} + 1231572270 p^{7} T^{22} - 114365838 p^{8} T^{23} + 12733121 p^{9} T^{24} - 995064 p^{10} T^{25} + 96745 p^{11} T^{26} - 5832 p^{12} T^{27} + 468 p^{13} T^{28} - 17 p^{14} T^{29} + p^{15} T^{30} \) | |
| 47 | \( 1 - 4 T + 413 T^{2} - 1442 T^{3} + 87710 T^{4} - 267304 T^{5} + 12508995 T^{6} - 33528668 T^{7} + 1332844440 T^{8} - 3168057464 T^{9} + 111985612082 T^{10} - 238244715368 T^{11} + 7647668213087 T^{12} - 14692261090038 T^{13} + 431642755789841 T^{14} - 754787470237648 T^{15} + 431642755789841 p T^{16} - 14692261090038 p^{2} T^{17} + 7647668213087 p^{3} T^{18} - 238244715368 p^{4} T^{19} + 111985612082 p^{5} T^{20} - 3168057464 p^{6} T^{21} + 1332844440 p^{7} T^{22} - 33528668 p^{8} T^{23} + 12508995 p^{9} T^{24} - 267304 p^{10} T^{25} + 87710 p^{11} T^{26} - 1442 p^{12} T^{27} + 413 p^{13} T^{28} - 4 p^{14} T^{29} + p^{15} T^{30} \) | |
| 53 | \( 1 - 30 T + 724 T^{2} - 12458 T^{3} + 182979 T^{4} - 2255320 T^{5} + 24477695 T^{6} - 232444258 T^{7} + 1956407158 T^{8} - 14335304202 T^{9} + 89772021115 T^{10} - 439121405762 T^{11} + 1217890348452 T^{12} + 5568715670600 T^{13} - 115901204908276 T^{14} + 1052506547263340 T^{15} - 115901204908276 p T^{16} + 5568715670600 p^{2} T^{17} + 1217890348452 p^{3} T^{18} - 439121405762 p^{4} T^{19} + 89772021115 p^{5} T^{20} - 14335304202 p^{6} T^{21} + 1956407158 p^{7} T^{22} - 232444258 p^{8} T^{23} + 24477695 p^{9} T^{24} - 2255320 p^{10} T^{25} + 182979 p^{11} T^{26} - 12458 p^{12} T^{27} + 724 p^{13} T^{28} - 30 p^{14} T^{29} + p^{15} T^{30} \) | |
| 59 | \( 1 - 10 T + 586 T^{2} - 5078 T^{3} + 161715 T^{4} - 1272114 T^{5} + 28592884 T^{6} - 211972356 T^{7} + 62604746 p T^{8} - 26417595966 T^{9} + 374458497537 T^{10} - 2598844430730 T^{11} + 31066713436820 T^{12} - 206888700598918 T^{13} + 2161091347919503 T^{14} - 13474170924206200 T^{15} + 2161091347919503 p T^{16} - 206888700598918 p^{2} T^{17} + 31066713436820 p^{3} T^{18} - 2598844430730 p^{4} T^{19} + 374458497537 p^{5} T^{20} - 26417595966 p^{6} T^{21} + 62604746 p^{8} T^{22} - 211972356 p^{8} T^{23} + 28592884 p^{9} T^{24} - 1272114 p^{10} T^{25} + 161715 p^{11} T^{26} - 5078 p^{12} T^{27} + 586 p^{13} T^{28} - 10 p^{14} T^{29} + p^{15} T^{30} \) | |
| 61 | \( 1 + 5 T + 546 T^{2} + 2710 T^{3} + 145651 T^{4} + 672638 T^{5} + 25043011 T^{6} + 102195914 T^{7} + 3105123638 T^{8} + 10698981425 T^{9} + 297465102859 T^{10} + 835597984462 T^{11} + 23336202137556 T^{12} + 53662507492692 T^{13} + 1583771261644834 T^{14} + 3243676240197092 T^{15} + 1583771261644834 p T^{16} + 53662507492692 p^{2} T^{17} + 23336202137556 p^{3} T^{18} + 835597984462 p^{4} T^{19} + 297465102859 p^{5} T^{20} + 10698981425 p^{6} T^{21} + 3105123638 p^{7} T^{22} + 102195914 p^{8} T^{23} + 25043011 p^{9} T^{24} + 672638 p^{10} T^{25} + 145651 p^{11} T^{26} + 2710 p^{12} T^{27} + 546 p^{13} T^{28} + 5 p^{14} T^{29} + p^{15} T^{30} \) | |
| 67 | \( 1 - 25 T + 893 T^{2} - 16763 T^{3} + 353650 T^{4} - 5407157 T^{5} + 1283326 p T^{6} - 1118792966 T^{7} + 14674986688 T^{8} - 166922705839 T^{9} + 1889517312756 T^{10} - 19118618636985 T^{11} + 191611851370023 T^{12} - 1743748863016707 T^{13} + 15698797503273223 T^{14} - 129204832608092908 T^{15} + 15698797503273223 p T^{16} - 1743748863016707 p^{2} T^{17} + 191611851370023 p^{3} T^{18} - 19118618636985 p^{4} T^{19} + 1889517312756 p^{5} T^{20} - 166922705839 p^{6} T^{21} + 14674986688 p^{7} T^{22} - 1118792966 p^{8} T^{23} + 1283326 p^{10} T^{24} - 5407157 p^{10} T^{25} + 353650 p^{11} T^{26} - 16763 p^{12} T^{27} + 893 p^{13} T^{28} - 25 p^{14} T^{29} + p^{15} T^{30} \) | |
| 71 | \( 1 - 44 T + 1378 T^{2} - 32034 T^{3} + 638448 T^{4} - 11028852 T^{5} + 172153587 T^{6} - 2442385684 T^{7} + 32061367795 T^{8} - 391116894108 T^{9} + 4473125554028 T^{10} - 48100727578814 T^{11} + 488527819464398 T^{12} - 4694967466974020 T^{13} + 42785914308357249 T^{14} - 370004356496963992 T^{15} + 42785914308357249 p T^{16} - 4694967466974020 p^{2} T^{17} + 488527819464398 p^{3} T^{18} - 48100727578814 p^{4} T^{19} + 4473125554028 p^{5} T^{20} - 391116894108 p^{6} T^{21} + 32061367795 p^{7} T^{22} - 2442385684 p^{8} T^{23} + 172153587 p^{9} T^{24} - 11028852 p^{10} T^{25} + 638448 p^{11} T^{26} - 32034 p^{12} T^{27} + 1378 p^{13} T^{28} - 44 p^{14} T^{29} + p^{15} T^{30} \) | |
| 73 | \( 1 - 11 T + 638 T^{2} - 5880 T^{3} + 199061 T^{4} - 1632268 T^{5} + 41521869 T^{6} - 312645340 T^{7} + 6522642852 T^{8} - 45697006359 T^{9} + 817272809187 T^{10} - 5346559976932 T^{11} + 84316831118676 T^{12} - 513674660671378 T^{13} + 7288713526297364 T^{14} - 41040685408969232 T^{15} + 7288713526297364 p T^{16} - 513674660671378 p^{2} T^{17} + 84316831118676 p^{3} T^{18} - 5346559976932 p^{4} T^{19} + 817272809187 p^{5} T^{20} - 45697006359 p^{6} T^{21} + 6522642852 p^{7} T^{22} - 312645340 p^{8} T^{23} + 41521869 p^{9} T^{24} - 1632268 p^{10} T^{25} + 199061 p^{11} T^{26} - 5880 p^{12} T^{27} + 638 p^{13} T^{28} - 11 p^{14} T^{29} + p^{15} T^{30} \) | |
| 79 | \( 1 - 3 T + 556 T^{2} - 544 T^{3} + 143437 T^{4} + 107246 T^{5} + 24098175 T^{6} + 40764098 T^{7} + 3145408524 T^{8} + 4880404749 T^{9} + 344331563863 T^{10} + 296813141206 T^{11} + 32010252646626 T^{12} + 12258803488132 T^{13} + 2635603093143690 T^{14} + 678265394788440 T^{15} + 2635603093143690 p T^{16} + 12258803488132 p^{2} T^{17} + 32010252646626 p^{3} T^{18} + 296813141206 p^{4} T^{19} + 344331563863 p^{5} T^{20} + 4880404749 p^{6} T^{21} + 3145408524 p^{7} T^{22} + 40764098 p^{8} T^{23} + 24098175 p^{9} T^{24} + 107246 p^{10} T^{25} + 143437 p^{11} T^{26} - 544 p^{12} T^{27} + 556 p^{13} T^{28} - 3 p^{14} T^{29} + p^{15} T^{30} \) | |
| 83 | \( 1 - 4 T + 782 T^{2} - 4554 T^{3} + 300279 T^{4} - 2156808 T^{5} + 76706005 T^{6} - 608091816 T^{7} + 14679644564 T^{8} - 118940055474 T^{9} + 2218690805423 T^{10} - 17502543979292 T^{11} + 271498571327028 T^{12} - 2022365493331522 T^{13} + 27266993023992286 T^{14} - 187092166973159140 T^{15} + 27266993023992286 p T^{16} - 2022365493331522 p^{2} T^{17} + 271498571327028 p^{3} T^{18} - 17502543979292 p^{4} T^{19} + 2218690805423 p^{5} T^{20} - 118940055474 p^{6} T^{21} + 14679644564 p^{7} T^{22} - 608091816 p^{8} T^{23} + 76706005 p^{9} T^{24} - 2156808 p^{10} T^{25} + 300279 p^{11} T^{26} - 4554 p^{12} T^{27} + 782 p^{13} T^{28} - 4 p^{14} T^{29} + p^{15} T^{30} \) | |
| 89 | \( 1 - T + 725 T^{2} - 891 T^{3} + 259844 T^{4} - 302395 T^{5} + 61304886 T^{6} - 52611244 T^{7} + 10723515602 T^{8} - 4039938985 T^{9} + 1491298621080 T^{10} + 240236023731 T^{11} + 173240650236995 T^{12} + 100308080988029 T^{13} + 17434205980178659 T^{14} + 12375337997967256 T^{15} + 17434205980178659 p T^{16} + 100308080988029 p^{2} T^{17} + 173240650236995 p^{3} T^{18} + 240236023731 p^{4} T^{19} + 1491298621080 p^{5} T^{20} - 4039938985 p^{6} T^{21} + 10723515602 p^{7} T^{22} - 52611244 p^{8} T^{23} + 61304886 p^{9} T^{24} - 302395 p^{10} T^{25} + 259844 p^{11} T^{26} - 891 p^{12} T^{27} + 725 p^{13} T^{28} - p^{14} T^{29} + p^{15} T^{30} \) | |
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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
−2.14650159062674668414357787428, −2.08524839219933263626335649112, −1.93759014596552205141629896736, −1.89168989050010141466826442772, −1.88221751490652709942186276069, −1.84367015945841941772073341729, −1.81995567885801100851814810568, −1.72262494198441577117858840362, −1.68756296187341538661712138608, −1.23843253135599203971357786321, −1.23254975519200215231702844093, −1.15952805072366031165959845904, −1.14670970041054360300477353007, −1.14473285732913068603940661984, −1.12368401461390670426017822612, −1.01997787984991228568661091809, −0.945635830395168938113505006246, −0.76962301468385557402147586130, −0.75694833546575663448607051476, −0.74728784040492522632582910957, −0.72771616411909823446222334602, −0.68840033838225526524925406182, −0.67972307577675164384469786224, −0.62333934425075659651833573526, −0.34788209365404458946201429466, 0.34788209365404458946201429466, 0.62333934425075659651833573526, 0.67972307577675164384469786224, 0.68840033838225526524925406182, 0.72771616411909823446222334602, 0.74728784040492522632582910957, 0.75694833546575663448607051476, 0.76962301468385557402147586130, 0.945635830395168938113505006246, 1.01997787984991228568661091809, 1.12368401461390670426017822612, 1.14473285732913068603940661984, 1.14670970041054360300477353007, 1.15952805072366031165959845904, 1.23254975519200215231702844093, 1.23843253135599203971357786321, 1.68756296187341538661712138608, 1.72262494198441577117858840362, 1.81995567885801100851814810568, 1.84367015945841941772073341729, 1.88221751490652709942186276069, 1.89168989050010141466826442772, 1.93759014596552205141629896736, 2.08524839219933263626335649112, 2.14650159062674668414357787428
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.