| L(s) = 1 | + 1.96·2-s + 3.37·3-s − 4.15·4-s + 14.4·5-s + 6.61·6-s + 7.73·7-s − 23.8·8-s − 15.6·9-s + 28.3·10-s − 22.3·11-s − 14.0·12-s + 16.3·13-s + 15.1·14-s + 48.6·15-s − 13.4·16-s − 116.·17-s − 30.6·18-s − 25.6·19-s − 60.0·20-s + 26.0·21-s − 43.8·22-s + 161.·23-s − 80.3·24-s + 83.4·25-s + 32.0·26-s − 143.·27-s − 32.1·28-s + ⋯ |
| L(s) = 1 | + 0.693·2-s + 0.648·3-s − 0.519·4-s + 1.29·5-s + 0.449·6-s + 0.417·7-s − 1.05·8-s − 0.578·9-s + 0.895·10-s − 0.613·11-s − 0.337·12-s + 0.348·13-s + 0.289·14-s + 0.838·15-s − 0.210·16-s − 1.66·17-s − 0.401·18-s − 0.309·19-s − 0.671·20-s + 0.270·21-s − 0.425·22-s + 1.46·23-s − 0.683·24-s + 0.667·25-s + 0.241·26-s − 1.02·27-s − 0.216·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.969700266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.969700266\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 37T \) |
| good | 2 | \( 1 - 1.96T + 8T^{2} \) |
| 3 | \( 1 - 3.37T + 27T^{2} \) |
| 5 | \( 1 - 14.4T + 125T^{2} \) |
| 7 | \( 1 - 7.73T + 343T^{2} \) |
| 11 | \( 1 + 22.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 25.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 100.T + 2.97e4T^{2} \) |
| 41 | \( 1 - 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 205.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 60.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 726.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 77.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 48.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 586.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 880.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 287.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 277.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 672.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 280.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.72e3T + 9.12e5T^{2} \) |
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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
−15.45688655072273386189213295591, −14.34155995971846300635778801818, −13.63690810944460456344707740065, −12.87639416227439566042142129215, −10.99615177881916166752336990091, −9.378793286570060407164660645748, −8.498498073222671690724363421669, −6.18335410772384154910714091429, −4.81312500994488368425117293648, −2.66543970646809800614893189240,
2.66543970646809800614893189240, 4.81312500994488368425117293648, 6.18335410772384154910714091429, 8.498498073222671690724363421669, 9.378793286570060407164660645748, 10.99615177881916166752336990091, 12.87639416227439566042142129215, 13.63690810944460456344707740065, 14.34155995971846300635778801818, 15.45688655072273386189213295591
