| L(s) = 1 | − 7.49e9·2-s − 4.87e15·3-s + 1.93e19·4-s − 6.28e22·5-s + 3.65e25·6-s − 4.41e27·7-s + 1.31e29·8-s + 1.34e31·9-s + 4.70e32·10-s + 2.47e33·11-s − 9.41e34·12-s + 8.18e35·13-s + 3.31e37·14-s + 3.06e38·15-s − 1.70e39·16-s + 1.52e40·17-s − 1.00e41·18-s − 1.62e41·19-s − 1.21e42·20-s + 2.15e43·21-s − 1.85e43·22-s − 1.77e44·23-s − 6.41e44·24-s + 1.23e45·25-s − 6.13e45·26-s − 1.53e46·27-s − 8.53e46·28-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 1.51·3-s + 0.523·4-s − 1.20·5-s + 1.87·6-s − 1.51·7-s + 0.587·8-s + 1.30·9-s + 1.48·10-s + 0.353·11-s − 0.795·12-s + 0.512·13-s + 1.86·14-s + 1.83·15-s − 1.24·16-s + 1.56·17-s − 1.61·18-s − 0.448·19-s − 0.631·20-s + 2.29·21-s − 0.435·22-s − 0.984·23-s − 0.892·24-s + 0.455·25-s − 0.633·26-s − 0.463·27-s − 0.791·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(66-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+65/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(33)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{67}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 7.49e9T + 3.68e19T^{2} \) |
| 3 | \( 1 + 4.87e15T + 1.03e31T^{2} \) |
| 5 | \( 1 + 6.28e22T + 2.71e45T^{2} \) |
| 7 | \( 1 + 4.41e27T + 8.53e54T^{2} \) |
| 11 | \( 1 - 2.47e33T + 4.90e67T^{2} \) |
| 13 | \( 1 - 8.18e35T + 2.54e72T^{2} \) |
| 17 | \( 1 - 1.52e40T + 9.53e79T^{2} \) |
| 19 | \( 1 + 1.62e41T + 1.31e83T^{2} \) |
| 23 | \( 1 + 1.77e44T + 3.25e88T^{2} \) |
| 29 | \( 1 + 4.19e47T + 1.13e95T^{2} \) |
| 31 | \( 1 + 6.78e47T + 8.67e96T^{2} \) |
| 37 | \( 1 - 1.36e51T + 8.57e101T^{2} \) |
| 41 | \( 1 - 3.18e52T + 6.77e104T^{2} \) |
| 43 | \( 1 - 1.25e53T + 1.49e106T^{2} \) |
| 47 | \( 1 + 9.94e53T + 4.85e108T^{2} \) |
| 53 | \( 1 + 1.30e55T + 1.19e112T^{2} \) |
| 59 | \( 1 + 1.98e57T + 1.27e115T^{2} \) |
| 61 | \( 1 + 5.55e57T + 1.11e116T^{2} \) |
| 67 | \( 1 - 1.23e59T + 4.95e118T^{2} \) |
| 71 | \( 1 + 5.36e59T + 2.14e120T^{2} \) |
| 73 | \( 1 - 5.21e60T + 1.30e121T^{2} \) |
| 79 | \( 1 - 5.19e61T + 2.21e123T^{2} \) |
| 83 | \( 1 - 8.74e60T + 5.49e124T^{2} \) |
| 89 | \( 1 + 1.81e63T + 5.13e126T^{2} \) |
| 97 | \( 1 + 1.86e63T + 1.38e129T^{2} \) |
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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
−16.72996491532779078496245999724, −16.07848675738605894542391583044, −12.47230606349043069152947573063, −11.07843726755068512700324839633, −9.672711911031898333862188867002, −7.62489546586294146271189935109, −6.10838062101252633943688578069, −3.91783513697314506031043975933, −0.836090781875320704259488231914, 0,
0.836090781875320704259488231914, 3.91783513697314506031043975933, 6.10838062101252633943688578069, 7.62489546586294146271189935109, 9.672711911031898333862188867002, 11.07843726755068512700324839633, 12.47230606349043069152947573063, 16.07848675738605894542391583044, 16.72996491532779078496245999724
