| L(s) = 1 | + 1.37e11·2-s − 8.29e17·3-s + 1.88e22·4-s − 2.22e26·5-s − 1.13e29·6-s − 1.55e31·7-s + 2.59e33·8-s + 7.96e34·9-s − 3.05e37·10-s + 1.87e39·11-s − 1.56e40·12-s + 9.98e41·13-s − 2.13e42·14-s + 1.84e44·15-s + 3.56e44·16-s + 4.07e44·17-s + 1.09e46·18-s − 1.17e48·19-s − 4.20e48·20-s + 1.28e49·21-s + 2.58e50·22-s + 6.36e50·23-s − 2.15e51·24-s + 2.30e52·25-s + 1.37e53·26-s + 4.38e53·27-s − 2.93e53·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.06·3-s + 0.5·4-s − 1.36·5-s − 0.751·6-s − 0.316·7-s + 0.353·8-s + 0.130·9-s − 0.967·10-s + 1.66·11-s − 0.531·12-s + 1.68·13-s − 0.223·14-s + 1.45·15-s + 0.250·16-s + 0.0293·17-s + 0.0925·18-s − 1.30·19-s − 0.684·20-s + 0.336·21-s + 1.17·22-s + 0.548·23-s − 0.375·24-s + 0.872·25-s + 1.19·26-s + 0.924·27-s − 0.158·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+75/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(38)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.37e11T \) |
| good | 3 | \( 1 + 8.29e17T + 6.08e35T^{2} \) |
| 5 | \( 1 + 2.22e26T + 2.64e52T^{2} \) |
| 7 | \( 1 + 1.55e31T + 2.41e63T^{2} \) |
| 11 | \( 1 - 1.87e39T + 1.27e78T^{2} \) |
| 13 | \( 1 - 9.98e41T + 3.51e83T^{2} \) |
| 17 | \( 1 - 4.07e44T + 1.92e92T^{2} \) |
| 19 | \( 1 + 1.17e48T + 8.06e95T^{2} \) |
| 23 | \( 1 - 6.36e50T + 1.34e102T^{2} \) |
| 29 | \( 1 + 6.06e54T + 4.78e109T^{2} \) |
| 31 | \( 1 + 8.48e55T + 7.11e111T^{2} \) |
| 37 | \( 1 - 4.74e58T + 4.12e117T^{2} \) |
| 41 | \( 1 + 1.53e60T + 9.09e120T^{2} \) |
| 43 | \( 1 + 7.65e60T + 3.23e122T^{2} \) |
| 47 | \( 1 - 5.36e62T + 2.55e125T^{2} \) |
| 53 | \( 1 + 3.23e64T + 2.09e129T^{2} \) |
| 59 | \( 1 + 1.88e66T + 6.51e132T^{2} \) |
| 61 | \( 1 - 1.66e67T + 7.93e133T^{2} \) |
| 67 | \( 1 - 4.39e68T + 9.02e136T^{2} \) |
| 71 | \( 1 + 4.24e69T + 6.98e138T^{2} \) |
| 73 | \( 1 + 8.11e69T + 5.61e139T^{2} \) |
| 79 | \( 1 + 1.61e71T + 2.09e142T^{2} \) |
| 83 | \( 1 + 6.54e71T + 8.52e143T^{2} \) |
| 89 | \( 1 + 1.38e73T + 1.60e146T^{2} \) |
| 97 | \( 1 - 5.00e74T + 1.01e149T^{2} \) |
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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
−12.81381007113353273853018461538, −11.56983793897731553732502485269, −11.06245184269766433204164575469, −8.640600425571915129463637726650, −6.84110979811310747507080577132, −5.93526480929049708299364385639, −4.25885723468124940412466794837, −3.53533031721212555383656345365, −1.26567291785491070935497903762, 0,
1.26567291785491070935497903762, 3.53533031721212555383656345365, 4.25885723468124940412466794837, 5.93526480929049708299364385639, 6.84110979811310747507080577132, 8.640600425571915129463637726650, 11.06245184269766433204164575469, 11.56983793897731553732502485269, 12.81381007113353273853018461538
