| L(s) = 1 | − 4.29e9·2-s − 2.22e15·3-s + 1.84e19·4-s + 2.44e22·5-s + 9.57e24·6-s + 8.86e26·7-s − 7.92e28·8-s − 5.33e30·9-s − 1.05e32·10-s − 1.95e33·11-s − 4.11e34·12-s − 3.37e35·13-s − 3.80e36·14-s − 5.45e37·15-s + 3.40e38·16-s − 8.16e38·17-s + 2.29e40·18-s + 4.91e41·19-s + 4.51e41·20-s − 1.97e42·21-s + 8.41e42·22-s + 2.14e43·23-s + 1.76e44·24-s − 2.11e45·25-s + 1.45e45·26-s + 3.48e46·27-s + 1.63e46·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.694·3-s + 0.5·4-s + 0.470·5-s + 0.491·6-s + 0.303·7-s − 0.353·8-s − 0.517·9-s − 0.332·10-s − 0.279·11-s − 0.347·12-s − 0.211·13-s − 0.214·14-s − 0.326·15-s + 0.250·16-s − 0.0836·17-s + 0.366·18-s + 1.35·19-s + 0.235·20-s − 0.210·21-s + 0.197·22-s + 0.119·23-s + 0.245·24-s − 0.778·25-s + 0.149·26-s + 1.05·27-s + 0.151·28-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
| \(L(33)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{67}{2})\) |
|
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 | \( 1 + 2.22e15T + 1.03e31T^{2} \) |
| 5 | \( 1 - 2.44e22T + 2.71e45T^{2} \) |
| 7 | \( 1 - 8.86e26T + 8.53e54T^{2} \) |
| 11 | \( 1 + 1.95e33T + 4.90e67T^{2} \) |
| 13 | \( 1 + 3.37e35T + 2.54e72T^{2} \) |
| 17 | \( 1 + 8.16e38T + 9.53e79T^{2} \) |
| 19 | \( 1 - 4.91e41T + 1.31e83T^{2} \) |
| 23 | \( 1 - 2.14e43T + 3.25e88T^{2} \) |
| 29 | \( 1 - 4.84e47T + 1.13e95T^{2} \) |
| 31 | \( 1 - 2.98e47T + 8.67e96T^{2} \) |
| 37 | \( 1 + 3.98e50T + 8.57e101T^{2} \) |
| 41 | \( 1 - 2.36e52T + 6.77e104T^{2} \) |
| 43 | \( 1 - 5.86e52T + 1.49e106T^{2} \) |
| 47 | \( 1 + 1.98e54T + 4.85e108T^{2} \) |
| 53 | \( 1 + 1.09e56T + 1.19e112T^{2} \) |
| 59 | \( 1 + 6.12e57T + 1.27e115T^{2} \) |
| 61 | \( 1 + 1.41e58T + 1.11e116T^{2} \) |
| 67 | \( 1 + 1.83e59T + 4.95e118T^{2} \) |
| 71 | \( 1 - 2.55e60T + 2.14e120T^{2} \) |
| 73 | \( 1 - 4.28e60T + 1.30e121T^{2} \) |
| 79 | \( 1 + 2.39e61T + 2.21e123T^{2} \) |
| 83 | \( 1 + 1.47e62T + 5.49e124T^{2} \) |
| 89 | \( 1 - 8.60e61T + 5.13e126T^{2} \) |
| 97 | \( 1 - 6.09e64T + 1.38e129T^{2} \) |
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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
−14.05038424983186729223448934002, −12.07418719110149735378284632565, −10.87509831698148236449499847472, −9.504235129703417686099242247066, −7.919141235857927421284308733405, −6.30210515639798906454265839639, −5.08416716072312763036874894869, −2.83751588423496613046798587259, −1.30009721069984967462026459068, 0,
1.30009721069984967462026459068, 2.83751588423496613046798587259, 5.08416716072312763036874894869, 6.30210515639798906454265839639, 7.919141235857927421284308733405, 9.504235129703417686099242247066, 10.87509831698148236449499847472, 12.07418719110149735378284632565, 14.05038424983186729223448934002
