| L(s) = 1 | − 6.87e10·2-s + 3.00e17·3-s + 4.72e21·4-s − 1.10e25·5-s − 2.06e28·6-s + 7.83e29·7-s − 3.24e32·8-s + 2.27e34·9-s + 7.58e35·10-s + 3.40e36·11-s + 1.41e39·12-s + 3.80e40·13-s − 5.38e40·14-s − 3.31e42·15-s + 2.23e43·16-s − 2.38e44·17-s − 1.56e45·18-s − 4.58e46·19-s − 5.21e46·20-s + 2.35e47·21-s − 2.34e47·22-s − 5.99e49·23-s − 9.75e49·24-s − 9.36e50·25-s − 2.61e51·26-s − 1.34e52·27-s + 3.69e51·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 0.5·4-s − 0.339·5-s − 0.817·6-s + 0.111·7-s − 0.353·8-s + 0.336·9-s + 0.239·10-s + 0.0332·11-s + 0.578·12-s + 0.835·13-s − 0.0789·14-s − 0.392·15-s + 0.250·16-s − 0.292·17-s − 0.238·18-s − 0.970·19-s − 0.169·20-s + 0.129·21-s − 0.0235·22-s − 1.18·23-s − 0.408·24-s − 0.884·25-s − 0.590·26-s − 0.766·27-s + 0.0558·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(74-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+73/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(37)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{75}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 6.87e10T \) |
| good | 3 | \( 1 - 3.00e17T + 6.75e34T^{2} \) |
| 5 | \( 1 + 1.10e25T + 1.05e51T^{2} \) |
| 7 | \( 1 - 7.83e29T + 4.92e61T^{2} \) |
| 11 | \( 1 - 3.40e36T + 1.05e76T^{2} \) |
| 13 | \( 1 - 3.80e40T + 2.07e81T^{2} \) |
| 17 | \( 1 + 2.38e44T + 6.64e89T^{2} \) |
| 19 | \( 1 + 4.58e46T + 2.23e93T^{2} \) |
| 23 | \( 1 + 5.99e49T + 2.54e99T^{2} \) |
| 29 | \( 1 - 3.66e53T + 5.68e106T^{2} \) |
| 31 | \( 1 - 1.75e54T + 7.40e108T^{2} \) |
| 37 | \( 1 - 1.64e57T + 3.01e114T^{2} \) |
| 41 | \( 1 - 1.95e58T + 5.41e117T^{2} \) |
| 43 | \( 1 + 7.77e59T + 1.75e119T^{2} \) |
| 47 | \( 1 + 5.34e60T + 1.15e122T^{2} \) |
| 53 | \( 1 - 5.17e62T + 7.44e125T^{2} \) |
| 59 | \( 1 + 1.88e63T + 1.87e129T^{2} \) |
| 61 | \( 1 - 1.49e65T + 2.13e130T^{2} \) |
| 67 | \( 1 + 8.11e66T + 2.01e133T^{2} \) |
| 71 | \( 1 + 2.29e67T + 1.38e135T^{2} \) |
| 73 | \( 1 + 1.30e68T + 1.05e136T^{2} \) |
| 79 | \( 1 + 7.20e67T + 3.36e138T^{2} \) |
| 83 | \( 1 + 1.10e70T + 1.23e140T^{2} \) |
| 89 | \( 1 - 1.46e71T + 2.02e142T^{2} \) |
| 97 | \( 1 - 7.46e71T + 1.08e145T^{2} \) |
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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
−13.48788706224126095897882173085, −11.64634428293895705985415551673, −10.07020783405444104817877758402, −8.631111554541227607800746801140, −7.988691924381378440584050687617, −6.31929144656032340805686992663, −4.08633000240039831521118812534, −2.78887457854937774929367771242, −1.60866412184828572701398358947, 0,
1.60866412184828572701398358947, 2.78887457854937774929367771242, 4.08633000240039831521118812534, 6.31929144656032340805686992663, 7.988691924381378440584050687617, 8.631111554541227607800746801140, 10.07020783405444104817877758402, 11.64634428293895705985415551673, 13.48788706224126095897882173085
