| L(s) = 1 | + 4-s + 20·5-s − 22·9-s + 2·16-s − 12·17-s + 20·20-s + 210·25-s − 22·36-s − 440·45-s + 70·49-s − 40·61-s − 2·64-s − 12·68-s − 36·73-s + 40·80-s + 229·81-s − 240·85-s + 210·100-s + 84·121-s + 1.54e3·125-s + 127-s + 131-s + 137-s + 139-s − 44·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 8.94·5-s − 7.33·9-s + 1/2·16-s − 2.91·17-s + 4.47·20-s + 42·25-s − 3.66·36-s − 65.5·45-s + 10·49-s − 5.12·61-s − 1/4·64-s − 1.45·68-s − 4.21·73-s + 4.47·80-s + 25.4·81-s − 26.0·85-s + 21·100-s + 7.63·121-s + 137.·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.66·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(54.64912319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(54.64912319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{2} - T^{4} + 5 T^{6} - p^{2} T^{8} - p^{3} T^{10} - p^{4} T^{12} + 5 p^{4} T^{14} - p^{6} T^{16} - p^{8} T^{18} + p^{10} T^{20} \) |
| 5 | \( ( 1 - T )^{20} \) |
| 19 | \( 1 - 46 T^{2} + 2229 T^{4} - 63720 T^{6} + 1756082 T^{8} - 34025364 T^{10} + 1756082 p^{2} T^{12} - 63720 p^{4} T^{14} + 2229 p^{6} T^{16} - 46 p^{8} T^{18} + p^{10} T^{20} \) |
| good | 3 | \( ( 1 + 11 T^{2} + 67 T^{4} + 310 T^{6} + 1228 T^{8} + 4078 T^{10} + 1228 p^{2} T^{12} + 310 p^{4} T^{14} + 67 p^{6} T^{16} + 11 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 7 | \( ( 1 - 5 p T^{2} + 13 p^{2} T^{4} - 7920 T^{6} + 75298 T^{8} - 580474 T^{10} + 75298 p^{2} T^{12} - 7920 p^{4} T^{14} + 13 p^{8} T^{16} - 5 p^{9} T^{18} + p^{10} T^{20} )^{2} \) |
| 11 | \( ( 1 - 42 T^{2} + 945 T^{4} - 15280 T^{6} + 17970 p T^{8} - 2242444 T^{10} + 17970 p^{3} T^{12} - 15280 p^{4} T^{14} + 945 p^{6} T^{16} - 42 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 13 | \( ( 1 - 81 T^{2} + 3123 T^{4} - 77318 T^{6} + 1406412 T^{8} - 20271362 T^{10} + 1406412 p^{2} T^{12} - 77318 p^{4} T^{14} + 3123 p^{6} T^{16} - 81 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 17 | \( ( 1 + 3 T + 29 T^{2} + 72 T^{3} + 482 T^{4} + 970 T^{5} + 482 p T^{6} + 72 p^{2} T^{7} + 29 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{4} \) |
| 23 | \( ( 1 - 5 p T^{2} + 243 p T^{4} - 145088 T^{6} + 2159274 T^{8} - 30327738 T^{10} + 2159274 p^{2} T^{12} - 145088 p^{4} T^{14} + 243 p^{7} T^{16} - 5 p^{9} T^{18} + p^{10} T^{20} )^{2} \) |
| 29 | \( ( 1 - 149 T^{2} + 11405 T^{4} - 579980 T^{6} + 22332090 T^{8} - 704203358 T^{10} + 22332090 p^{2} T^{12} - 579980 p^{4} T^{14} + 11405 p^{6} T^{16} - 149 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 31 | \( ( 1 + 118 T^{2} + 8549 T^{4} + 464312 T^{6} + 19615962 T^{8} + 671931556 T^{10} + 19615962 p^{2} T^{12} + 464312 p^{4} T^{14} + 8549 p^{6} T^{16} + 118 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 37 | \( ( 1 - 90 T^{2} + 5591 T^{4} - 274716 T^{6} + 10833648 T^{8} - 410662612 T^{10} + 10833648 p^{2} T^{12} - 274716 p^{4} T^{14} + 5591 p^{6} T^{16} - 90 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 41 | \( ( 1 - 162 T^{2} + 14213 T^{4} - 846280 T^{6} + 39887114 T^{8} - 1667309740 T^{10} + 39887114 p^{2} T^{12} - 846280 p^{4} T^{14} + 14213 p^{6} T^{16} - 162 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 43 | \( ( 1 - 262 T^{2} + 35489 T^{4} - 3167248 T^{6} + 205476662 T^{8} - 10088194580 T^{10} + 205476662 p^{2} T^{12} - 3167248 p^{4} T^{14} + 35489 p^{6} T^{16} - 262 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 47 | \( ( 1 - 150 T^{2} + 14985 T^{4} - 1149936 T^{6} + 70507094 T^{8} - 3635903348 T^{10} + 70507094 p^{2} T^{12} - 1149936 p^{4} T^{14} + 14985 p^{6} T^{16} - 150 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 53 | \( ( 1 - 273 T^{2} + 36547 T^{4} - 3340870 T^{6} + 240374892 T^{8} - 14112838402 T^{10} + 240374892 p^{2} T^{12} - 3340870 p^{4} T^{14} + 36547 p^{6} T^{16} - 273 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 59 | \( ( 1 + 131 T^{2} + 6821 T^{4} + 32572 T^{6} - 11713966 T^{8} - 798731550 T^{10} - 11713966 p^{2} T^{12} + 32572 p^{4} T^{14} + 6821 p^{6} T^{16} + 131 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 61 | \( ( 1 + 10 T + 199 T^{2} + 1180 T^{3} + 15532 T^{4} + 69556 T^{5} + 15532 p T^{6} + 1180 p^{2} T^{7} + 199 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} )^{4} \) |
| 67 | \( ( 1 + 427 T^{2} + 93787 T^{4} + 13482806 T^{6} + 1392707764 T^{8} + 107369893678 T^{10} + 1392707764 p^{2} T^{12} + 13482806 p^{4} T^{14} + 93787 p^{6} T^{16} + 427 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 71 | \( ( 1 + 470 T^{2} + 107413 T^{4} + 15893048 T^{6} + 1697675034 T^{8} + 137445687652 T^{10} + 1697675034 p^{2} T^{12} + 15893048 p^{4} T^{14} + 107413 p^{6} T^{16} + 470 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 73 | \( ( 1 + 9 T + 253 T^{2} + 1940 T^{3} + 29162 T^{4} + 187430 T^{5} + 29162 p T^{6} + 1940 p^{2} T^{7} + 253 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{4} \) |
| 79 | \( ( 1 + 586 T^{2} + 167781 T^{4} + 30448712 T^{6} + 3857009658 T^{8} + 355227136476 T^{10} + 3857009658 p^{2} T^{12} + 30448712 p^{4} T^{14} + 167781 p^{6} T^{16} + 586 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 83 | \( ( 1 - 522 T^{2} + 136161 T^{4} - 23257648 T^{6} + 2889065766 T^{8} - 273059517772 T^{10} + 2889065766 p^{2} T^{12} - 23257648 p^{4} T^{14} + 136161 p^{6} T^{16} - 522 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 89 | \( ( 1 - 326 T^{2} + 64317 T^{4} - 9204360 T^{6} + 1054122674 T^{8} - 102207745764 T^{10} + 1054122674 p^{2} T^{12} - 9204360 p^{4} T^{14} + 64317 p^{6} T^{16} - 326 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 97 | \( ( 1 - 386 T^{2} + 96831 T^{4} - 16494732 T^{6} + 2240176928 T^{8} - 238521810852 T^{10} + 2240176928 p^{2} T^{12} - 16494732 p^{4} T^{14} + 96831 p^{6} T^{16} - 386 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.76884996334257673170135168865, −2.66814384243600396486244499568, −2.55686421781178560236475896137, −2.48614457997476788046996930860, −2.41863734802611702027554644477, −2.31063487523113169290668174936, −2.25630259884644626054458189413, −2.15397031693656120665401803279, −2.13005496437079041561960577384, −2.01684858867282915866874148720, −2.01305368551416263942928601030, −1.89362198653022054353598909960, −1.81222483636123119370992276408, −1.75214653268319458061708291799, −1.72711849723984562120126591188, −1.68138436935960184828571694948, −1.61694331285958950079605824750, −1.42017382956006588242094971389, −1.03686342966186611495152980464, −1.02305272815353583909409384473, −0.990658244188030031351196935459, −0.872223129172824655085976150220, −0.57757957547076359523170597984, −0.46671465769102662669032325787, −0.38702782785590867710455327163,
0.38702782785590867710455327163, 0.46671465769102662669032325787, 0.57757957547076359523170597984, 0.872223129172824655085976150220, 0.990658244188030031351196935459, 1.02305272815353583909409384473, 1.03686342966186611495152980464, 1.42017382956006588242094971389, 1.61694331285958950079605824750, 1.68138436935960184828571694948, 1.72711849723984562120126591188, 1.75214653268319458061708291799, 1.81222483636123119370992276408, 1.89362198653022054353598909960, 2.01305368551416263942928601030, 2.01684858867282915866874148720, 2.13005496437079041561960577384, 2.15397031693656120665401803279, 2.25630259884644626054458189413, 2.31063487523113169290668174936, 2.41863734802611702027554644477, 2.48614457997476788046996930860, 2.55686421781178560236475896137, 2.66814384243600396486244499568, 2.76884996334257673170135168865
Plot not available for L-functions of degree greater than 10.
