| L(s) = 1 | + 5.49e11·2-s − 8.21e18·3-s + 3.02e23·4-s − 6.30e27·5-s − 4.51e30·6-s + 3.82e33·7-s + 1.66e35·8-s + 1.81e37·9-s − 3.46e39·10-s − 8.25e40·11-s − 2.48e42·12-s − 6.74e43·13-s + 2.10e45·14-s + 5.17e46·15-s + 9.13e46·16-s + 2.03e48·17-s + 9.99e48·18-s + 3.74e50·19-s − 1.90e51·20-s − 3.14e52·21-s − 4.53e52·22-s + 8.53e53·23-s − 1.36e54·24-s + 2.31e55·25-s − 3.70e55·26-s + 2.55e56·27-s + 1.15e57·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.17·3-s + 0.5·4-s − 1.54·5-s − 0.827·6-s + 1.59·7-s + 0.353·8-s + 0.369·9-s − 1.09·10-s − 0.604·11-s − 0.585·12-s − 0.673·13-s + 1.12·14-s + 1.81·15-s + 0.250·16-s + 0.507·17-s + 0.260·18-s + 1.15·19-s − 0.774·20-s − 1.86·21-s − 0.427·22-s + 1.39·23-s − 0.413·24-s + 1.40·25-s − 0.476·26-s + 0.738·27-s + 0.795·28-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 | \( 1 + 8.21e18T + 4.92e37T^{2} \) |
| 5 | \( 1 + 6.30e27T + 1.65e55T^{2} \) |
| 7 | \( 1 - 3.82e33T + 5.79e66T^{2} \) |
| 11 | \( 1 + 8.25e40T + 1.86e82T^{2} \) |
| 13 | \( 1 + 6.74e43T + 1.00e88T^{2} \) |
| 17 | \( 1 - 2.03e48T + 1.60e97T^{2} \) |
| 19 | \( 1 - 3.74e50T + 1.05e101T^{2} \) |
| 23 | \( 1 - 8.53e53T + 3.77e107T^{2} \) |
| 29 | \( 1 - 7.45e57T + 3.38e115T^{2} \) |
| 31 | \( 1 + 1.19e59T + 6.57e117T^{2} \) |
| 37 | \( 1 + 9.28e61T + 7.72e123T^{2} \) |
| 41 | \( 1 - 2.58e63T + 2.56e127T^{2} \) |
| 43 | \( 1 + 5.39e64T + 1.10e129T^{2} \) |
| 47 | \( 1 + 1.50e66T + 1.24e132T^{2} \) |
| 53 | \( 1 + 5.02e67T + 1.65e136T^{2} \) |
| 59 | \( 1 + 1.39e70T + 7.89e139T^{2} \) |
| 61 | \( 1 + 3.14e70T + 1.09e141T^{2} \) |
| 67 | \( 1 - 8.22e71T + 1.81e144T^{2} \) |
| 71 | \( 1 - 7.87e72T + 1.77e146T^{2} \) |
| 73 | \( 1 - 4.25e73T + 1.59e147T^{2} \) |
| 79 | \( 1 + 3.92e74T + 8.17e149T^{2} \) |
| 83 | \( 1 + 4.63e75T + 4.04e151T^{2} \) |
| 89 | \( 1 + 5.55e76T + 1.00e154T^{2} \) |
| 97 | \( 1 + 3.14e78T + 9.01e156T^{2} \) |
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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
−12.32955254044194265661517998932, −11.55094817692017061353646919826, −10.84287122125300077107059439218, −8.091807527277091461251136934869, −7.13812590924770087482111290880, −5.16222825899425961174652250122, −4.80861865193965941424379036238, −3.19847960954068489244977756821, −1.25419273458959932703460050980, 0,
1.25419273458959932703460050980, 3.19847960954068489244977756821, 4.80861865193965941424379036238, 5.16222825899425961174652250122, 7.13812590924770087482111290880, 8.091807527277091461251136934869, 10.84287122125300077107059439218, 11.55094817692017061353646919826, 12.32955254044194265661517998932
