| L(s) = 1 | − 4.24·2-s + 3·3-s + 9.99·4-s − 12.7·6-s + 20.9·7-s − 8.48·8-s + 9·9-s + 16.0·11-s + 29.9·12-s + 34.9·13-s − 88.9·14-s − 44.0·16-s + 17·17-s − 38.1·18-s − 80.8·19-s + 62.9·21-s − 68.0·22-s + 115.·23-s − 25.4·24-s − 148.·26-s + 27·27-s + 209.·28-s + 154.·29-s + 299.·31-s + 254.·32-s + 48.0·33-s − 72.1·34-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 0.577·3-s + 1.24·4-s − 0.866·6-s + 1.13·7-s − 0.374·8-s + 0.333·9-s + 0.439·11-s + 0.721·12-s + 0.745·13-s − 1.69·14-s − 0.687·16-s + 0.242·17-s − 0.500·18-s − 0.976·19-s + 0.653·21-s − 0.659·22-s + 1.05·23-s − 0.216·24-s − 1.11·26-s + 0.192·27-s + 1.41·28-s + 0.986·29-s + 1.73·31-s + 1.40·32-s + 0.253·33-s − 0.363·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.678641553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678641553\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 4.24T + 8T^{2} \) |
| 7 | \( 1 - 20.9T + 343T^{2} \) |
| 11 | \( 1 - 16.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.9T + 2.19e3T^{2} \) |
| 19 | \( 1 + 80.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 299.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 315.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 132.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 23.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 260.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 629.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 461.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 789.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 686.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 484.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 254T + 4.93e5T^{2} \) |
| 83 | \( 1 + 548.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 925.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 732.T + 9.12e5T^{2} \) |
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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
−9.003535998651876752019289985465, −8.516287773935220348945673348710, −8.107784427391948959391271198791, −7.12978735617992524184432187813, −6.41880477895644631301898358519, −4.98333828728322345248571852349, −4.08060461168740891044659634658, −2.68299052140633936988185510876, −1.62132933971087292184976279183, −0.879154950982805729397334529224,
0.879154950982805729397334529224, 1.62132933971087292184976279183, 2.68299052140633936988185510876, 4.08060461168740891044659634658, 4.98333828728322345248571852349, 6.41880477895644631301898358519, 7.12978735617992524184432187813, 8.107784427391948959391271198791, 8.516287773935220348945673348710, 9.003535998651876752019289985465
