| L(s) = 1 | − 0.403·2-s + 1.58·3-s − 1.83·4-s − 5-s − 0.637·6-s + 3.59·7-s + 1.54·8-s − 0.502·9-s + 0.403·10-s + 11-s − 2.90·12-s − 2.34·13-s − 1.44·14-s − 1.58·15-s + 3.04·16-s − 1.11·17-s + 0.202·18-s + 3.68·19-s + 1.83·20-s + 5.67·21-s − 0.403·22-s − 8.55·23-s + 2.44·24-s + 25-s + 0.946·26-s − 5.53·27-s − 6.59·28-s + ⋯ |
| L(s) = 1 | − 0.285·2-s + 0.912·3-s − 0.918·4-s − 0.447·5-s − 0.260·6-s + 1.35·7-s + 0.547·8-s − 0.167·9-s + 0.127·10-s + 0.301·11-s − 0.838·12-s − 0.650·13-s − 0.387·14-s − 0.408·15-s + 0.762·16-s − 0.271·17-s + 0.0478·18-s + 0.845·19-s + 0.410·20-s + 1.23·21-s − 0.0860·22-s − 1.78·23-s + 0.499·24-s + 0.200·25-s + 0.185·26-s − 1.06·27-s − 1.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 0.403T + 2T^{2} \) |
| 3 | \( 1 - 1.58T + 3T^{2} \) |
| 7 | \( 1 - 3.59T + 7T^{2} \) |
| 13 | \( 1 + 2.34T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 - 3.68T + 19T^{2} \) |
| 23 | \( 1 + 8.55T + 23T^{2} \) |
| 29 | \( 1 + 7.37T + 29T^{2} \) |
| 31 | \( 1 - 1.70T + 31T^{2} \) |
| 37 | \( 1 + 6.91T + 37T^{2} \) |
| 41 | \( 1 + 2.30T + 41T^{2} \) |
| 43 | \( 1 - 2.37T + 43T^{2} \) |
| 47 | \( 1 + 9.18T + 47T^{2} \) |
| 53 | \( 1 + 0.155T + 53T^{2} \) |
| 59 | \( 1 - 8.63T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 0.586T + 71T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 2.90T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 19.5T + 97T^{2} \) |
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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
−8.185787266518418282180252596158, −7.77720017855408655443497213464, −6.94438362355669385702746940884, −5.52398163673507659650047050124, −5.08053303085330685749891942197, −4.06110030145103038612679171985, −3.64765717971376346820091862813, −2.35443386779844717483486777026, −1.49756873780806895807372729602, 0,
1.49756873780806895807372729602, 2.35443386779844717483486777026, 3.64765717971376346820091862813, 4.06110030145103038612679171985, 5.08053303085330685749891942197, 5.52398163673507659650047050124, 6.94438362355669385702746940884, 7.77720017855408655443497213464, 8.185787266518418282180252596158
