| L(s) = 1 | + 5.49e11·2-s + 7.60e18·3-s + 3.02e23·4-s − 2.65e27·5-s + 4.18e30·6-s − 1.85e33·7-s + 1.66e35·8-s + 8.58e36·9-s − 1.45e39·10-s + 1.28e40·11-s + 2.29e42·12-s + 9.72e42·13-s − 1.02e45·14-s − 2.01e46·15-s + 9.13e46·16-s + 5.80e48·17-s + 4.72e48·18-s − 1.76e50·19-s − 8.01e50·20-s − 1.41e52·21-s + 7.04e51·22-s − 3.63e53·23-s + 1.26e54·24-s − 9.51e54·25-s + 5.34e54·26-s − 3.09e56·27-s − 5.61e56·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.08·3-s + 0.5·4-s − 0.651·5-s + 0.766·6-s − 0.771·7-s + 0.353·8-s + 0.174·9-s − 0.460·10-s + 0.0938·11-s + 0.541·12-s + 0.0970·13-s − 0.545·14-s − 0.706·15-s + 0.250·16-s + 1.44·17-s + 0.123·18-s − 0.545·19-s − 0.325·20-s − 0.836·21-s + 0.0663·22-s − 0.592·23-s + 0.383·24-s − 0.575·25-s + 0.0686·26-s − 0.894·27-s − 0.385·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(80-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+79/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(40)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{81}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.49e11T \) |
| good | 3 | \( 1 - 7.60e18T + 4.92e37T^{2} \) |
| 5 | \( 1 + 2.65e27T + 1.65e55T^{2} \) |
| 7 | \( 1 + 1.85e33T + 5.79e66T^{2} \) |
| 11 | \( 1 - 1.28e40T + 1.86e82T^{2} \) |
| 13 | \( 1 - 9.72e42T + 1.00e88T^{2} \) |
| 17 | \( 1 - 5.80e48T + 1.60e97T^{2} \) |
| 19 | \( 1 + 1.76e50T + 1.05e101T^{2} \) |
| 23 | \( 1 + 3.63e53T + 3.77e107T^{2} \) |
| 29 | \( 1 + 6.59e57T + 3.38e115T^{2} \) |
| 31 | \( 1 + 5.88e58T + 6.57e117T^{2} \) |
| 37 | \( 1 + 9.06e61T + 7.72e123T^{2} \) |
| 41 | \( 1 + 9.50e63T + 2.56e127T^{2} \) |
| 43 | \( 1 - 2.02e64T + 1.10e129T^{2} \) |
| 47 | \( 1 + 1.19e66T + 1.24e132T^{2} \) |
| 53 | \( 1 - 2.24e68T + 1.65e136T^{2} \) |
| 59 | \( 1 - 4.05e69T + 7.89e139T^{2} \) |
| 61 | \( 1 - 2.62e70T + 1.09e141T^{2} \) |
| 67 | \( 1 - 3.14e71T + 1.81e144T^{2} \) |
| 71 | \( 1 + 1.14e73T + 1.77e146T^{2} \) |
| 73 | \( 1 + 2.75e73T + 1.59e147T^{2} \) |
| 79 | \( 1 - 9.08e74T + 8.17e149T^{2} \) |
| 83 | \( 1 + 1.14e76T + 4.04e151T^{2} \) |
| 89 | \( 1 + 1.62e77T + 1.00e154T^{2} \) |
| 97 | \( 1 - 1.53e78T + 9.01e156T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.03862951129368010732796319390, −11.74397567722722779603721845452, −9.938694593383628598907820005713, −8.425256241073473674646002341057, −7.25231175287166635444336440548, −5.66077261642081261445923183424, −3.81949989704460253595283965963, −3.25748674149963422995554692464, −1.88186552337273229405267214159, 0,
1.88186552337273229405267214159, 3.25748674149963422995554692464, 3.81949989704460253595283965963, 5.66077261642081261445923183424, 7.25231175287166635444336440548, 8.425256241073473674646002341057, 9.938694593383628598907820005713, 11.74397567722722779603721845452, 13.03862951129368010732796319390
