| L(s) = 1 | + 4.29e9·2-s − 2.64e15·3-s + 1.84e19·4-s − 6.66e22·5-s − 1.13e25·6-s + 1.54e27·7-s + 7.92e28·8-s − 3.29e30·9-s − 2.86e32·10-s − 6.12e33·11-s − 4.88e34·12-s − 1.65e35·13-s + 6.63e36·14-s + 1.76e38·15-s + 3.40e38·16-s − 1.55e40·17-s − 1.41e40·18-s − 7.97e40·19-s − 1.23e42·20-s − 4.09e42·21-s − 2.62e43·22-s + 2.69e44·23-s − 2.09e44·24-s + 1.73e45·25-s − 7.12e44·26-s + 3.59e46·27-s + 2.85e46·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.824·3-s + 0.5·4-s − 1.28·5-s − 0.583·6-s + 0.528·7-s + 0.353·8-s − 0.319·9-s − 0.905·10-s − 0.874·11-s − 0.412·12-s − 0.103·13-s + 0.373·14-s + 1.05·15-s + 0.250·16-s − 1.59·17-s − 0.226·18-s − 0.219·19-s − 0.640·20-s − 0.436·21-s − 0.618·22-s + 1.49·23-s − 0.291·24-s + 0.640·25-s − 0.0734·26-s + 1.08·27-s + 0.264·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(66-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+65/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(33)\) |
\(\approx\) |
\(1.284332430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.284332430\) |
| \(L(\frac{67}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.29e9T \) |
| good | 3 | \( 1 + 2.64e15T + 1.03e31T^{2} \) |
| 5 | \( 1 + 6.66e22T + 2.71e45T^{2} \) |
| 7 | \( 1 - 1.54e27T + 8.53e54T^{2} \) |
| 11 | \( 1 + 6.12e33T + 4.90e67T^{2} \) |
| 13 | \( 1 + 1.65e35T + 2.54e72T^{2} \) |
| 17 | \( 1 + 1.55e40T + 9.53e79T^{2} \) |
| 19 | \( 1 + 7.97e40T + 1.31e83T^{2} \) |
| 23 | \( 1 - 2.69e44T + 3.25e88T^{2} \) |
| 29 | \( 1 + 3.43e47T + 1.13e95T^{2} \) |
| 31 | \( 1 - 4.83e48T + 8.67e96T^{2} \) |
| 37 | \( 1 + 4.98e49T + 8.57e101T^{2} \) |
| 41 | \( 1 - 3.20e52T + 6.77e104T^{2} \) |
| 43 | \( 1 - 4.40e52T + 1.49e106T^{2} \) |
| 47 | \( 1 + 1.74e54T + 4.85e108T^{2} \) |
| 53 | \( 1 - 1.18e56T + 1.19e112T^{2} \) |
| 59 | \( 1 - 5.93e57T + 1.27e115T^{2} \) |
| 61 | \( 1 - 1.70e58T + 1.11e116T^{2} \) |
| 67 | \( 1 - 2.38e59T + 4.95e118T^{2} \) |
| 71 | \( 1 + 7.47e59T + 2.14e120T^{2} \) |
| 73 | \( 1 + 2.25e60T + 1.30e121T^{2} \) |
| 79 | \( 1 + 7.12e60T + 2.21e123T^{2} \) |
| 83 | \( 1 - 2.80e62T + 5.49e124T^{2} \) |
| 89 | \( 1 + 3.82e63T + 5.13e126T^{2} \) |
| 97 | \( 1 + 2.41e64T + 1.38e129T^{2} \) |
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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
−14.94181368230366433452560956629, −13.02262151801943087091315352467, −11.56907588211655378771318217010, −10.96650291763257909335616401582, −8.296716612835563553707527050017, −6.87424975320932289179146394437, −5.25665256081847827010448130546, −4.25521628636417732858790618043, −2.62427251309994093968122992386, −0.56747129256755184420359215793,
0.56747129256755184420359215793, 2.62427251309994093968122992386, 4.25521628636417732858790618043, 5.25665256081847827010448130546, 6.87424975320932289179146394437, 8.296716612835563553707527050017, 10.96650291763257909335616401582, 11.56907588211655378771318217010, 13.02262151801943087091315352467, 14.94181368230366433452560956629
