| L(s) = 1 | − 2.65·2-s + 3-s + 5.05·4-s − 1.68·5-s − 2.65·6-s − 3.96·7-s − 8.10·8-s + 9-s + 4.46·10-s − 5.81·11-s + 5.05·12-s − 13-s + 10.5·14-s − 1.68·15-s + 11.4·16-s − 3.34·17-s − 2.65·18-s − 4.92·19-s − 8.49·20-s − 3.96·21-s + 15.4·22-s + 2.84·23-s − 8.10·24-s − 2.17·25-s + 2.65·26-s + 27-s − 20.0·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.577·3-s + 2.52·4-s − 0.752·5-s − 1.08·6-s − 1.49·7-s − 2.86·8-s + 0.333·9-s + 1.41·10-s − 1.75·11-s + 1.45·12-s − 0.277·13-s + 2.81·14-s − 0.434·15-s + 2.85·16-s − 0.812·17-s − 0.625·18-s − 1.12·19-s − 1.89·20-s − 0.865·21-s + 3.29·22-s + 0.593·23-s − 1.65·24-s − 0.434·25-s + 0.520·26-s + 0.192·27-s − 3.78·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02355375871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02355375871\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 5 | \( 1 + 1.68T + 5T^{2} \) |
| 7 | \( 1 + 3.96T + 7T^{2} \) |
| 11 | \( 1 + 5.81T + 11T^{2} \) |
| 17 | \( 1 + 3.34T + 17T^{2} \) |
| 19 | \( 1 + 4.92T + 19T^{2} \) |
| 23 | \( 1 - 2.84T + 23T^{2} \) |
| 29 | \( 1 + 9.70T + 29T^{2} \) |
| 31 | \( 1 - 8.23T + 31T^{2} \) |
| 37 | \( 1 - 1.17T + 37T^{2} \) |
| 41 | \( 1 + 4.01T + 41T^{2} \) |
| 43 | \( 1 + 5.10T + 43T^{2} \) |
| 47 | \( 1 + 0.541T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 3.30T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 9.72T + 73T^{2} \) |
| 79 | \( 1 + 9.63T + 79T^{2} \) |
| 83 | \( 1 - 7.71T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−8.482850180120457595052509456237, −7.82094789446621732475279446422, −7.37675618924954082268467625775, −6.61244030194903769409165928676, −5.95508932090241251733518543468, −4.56033152793141801761362989410, −3.27465425812535102740938549108, −2.77567706635855836943254445556, −1.89535848054389295606961809799, −0.10351308079567085207504868585,
0.10351308079567085207504868585, 1.89535848054389295606961809799, 2.77567706635855836943254445556, 3.27465425812535102740938549108, 4.56033152793141801761362989410, 5.95508932090241251733518543468, 6.61244030194903769409165928676, 7.37675618924954082268467625775, 7.82094789446621732475279446422, 8.482850180120457595052509456237
