| L(s) = 1 | + 7.69e9·2-s − 5.56e15·3-s + 2.23e19·4-s + 6.36e22·5-s − 4.27e25·6-s + 1.82e27·7-s − 1.12e29·8-s + 2.06e31·9-s + 4.89e32·10-s − 6.60e31·11-s − 1.24e35·12-s − 2.71e36·13-s + 1.40e37·14-s − 3.54e38·15-s − 1.68e39·16-s + 2.66e39·17-s + 1.58e41·18-s − 5.18e41·19-s + 1.42e42·20-s − 1.01e43·21-s − 5.08e41·22-s − 9.69e43·23-s + 6.24e44·24-s + 1.34e45·25-s − 2.08e46·26-s − 5.75e46·27-s + 4.07e46·28-s + ⋯ |
| L(s) = 1 | + 1.26·2-s − 1.73·3-s + 0.604·4-s + 1.22·5-s − 2.19·6-s + 0.625·7-s − 0.500·8-s + 2.00·9-s + 1.54·10-s − 0.00943·11-s − 1.04·12-s − 1.70·13-s + 0.792·14-s − 2.11·15-s − 1.23·16-s + 0.273·17-s + 2.53·18-s − 1.42·19-s + 0.739·20-s − 1.08·21-s − 0.0119·22-s − 0.537·23-s + 0.867·24-s + 0.495·25-s − 2.15·26-s − 1.73·27-s + 0.378·28-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 | \( 1 - 7.69e9T + 3.68e19T^{2} \) |
| 3 | \( 1 + 5.56e15T + 1.03e31T^{2} \) |
| 5 | \( 1 - 6.36e22T + 2.71e45T^{2} \) |
| 7 | \( 1 - 1.82e27T + 8.53e54T^{2} \) |
| 11 | \( 1 + 6.60e31T + 4.90e67T^{2} \) |
| 13 | \( 1 + 2.71e36T + 2.54e72T^{2} \) |
| 17 | \( 1 - 2.66e39T + 9.53e79T^{2} \) |
| 19 | \( 1 + 5.18e41T + 1.31e83T^{2} \) |
| 23 | \( 1 + 9.69e43T + 3.25e88T^{2} \) |
| 29 | \( 1 - 8.81e46T + 1.13e95T^{2} \) |
| 31 | \( 1 + 1.18e48T + 8.67e96T^{2} \) |
| 37 | \( 1 - 4.45e50T + 8.57e101T^{2} \) |
| 41 | \( 1 + 1.11e52T + 6.77e104T^{2} \) |
| 43 | \( 1 + 1.95e53T + 1.49e106T^{2} \) |
| 47 | \( 1 + 1.87e54T + 4.85e108T^{2} \) |
| 53 | \( 1 - 1.20e56T + 1.19e112T^{2} \) |
| 59 | \( 1 - 1.90e57T + 1.27e115T^{2} \) |
| 61 | \( 1 + 3.82e57T + 1.11e116T^{2} \) |
| 67 | \( 1 - 2.48e59T + 4.95e118T^{2} \) |
| 71 | \( 1 + 8.10e59T + 2.14e120T^{2} \) |
| 73 | \( 1 + 2.44e60T + 1.30e121T^{2} \) |
| 79 | \( 1 + 5.39e61T + 2.21e123T^{2} \) |
| 83 | \( 1 - 2.95e62T + 5.49e124T^{2} \) |
| 89 | \( 1 - 1.30e63T + 5.13e126T^{2} \) |
| 97 | \( 1 + 1.88e64T + 1.38e129T^{2} \) |
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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
−16.94563206846050760421559306164, −14.68605716554908546315054734312, −12.95505916764587740938839241400, −11.77804973629832928600040167966, −10.08838205751244337962930170840, −6.54376882672648155736900490075, −5.43710714778185431817529393363, −4.62950461753859644043500508645, −2.01578865589714756326541336873, 0,
2.01578865589714756326541336873, 4.62950461753859644043500508645, 5.43710714778185431817529393363, 6.54376882672648155736900490075, 10.08838205751244337962930170840, 11.77804973629832928600040167966, 12.95505916764587740938839241400, 14.68605716554908546315054734312, 16.94563206846050760421559306164
