| L(s) = 1 | + 3.35·3-s + 5-s + 1.13·7-s + 8.25·9-s − 3.07·11-s + 5.67·13-s + 3.35·15-s + 7.16·17-s + 3.16·19-s + 3.79·21-s + 3.05·23-s + 25-s + 17.6·27-s − 4.42·29-s − 10.2·31-s − 10.3·33-s + 1.13·35-s − 1.76·37-s + 19.0·39-s − 11.6·41-s − 2.66·43-s + 8.25·45-s + 7.89·47-s − 5.72·49-s + 24.0·51-s − 6.92·53-s − 3.07·55-s + ⋯ |
| L(s) = 1 | + 1.93·3-s + 0.447·5-s + 0.427·7-s + 2.75·9-s − 0.926·11-s + 1.57·13-s + 0.866·15-s + 1.73·17-s + 0.725·19-s + 0.827·21-s + 0.637·23-s + 0.200·25-s + 3.39·27-s − 0.822·29-s − 1.84·31-s − 1.79·33-s + 0.191·35-s − 0.290·37-s + 3.04·39-s − 1.82·41-s − 0.405·43-s + 1.23·45-s + 1.15·47-s − 0.817·49-s + 3.36·51-s − 0.951·53-s − 0.414·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.974766911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.974766911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 3.35T + 3T^{2} \) |
| 7 | \( 1 - 1.13T + 7T^{2} \) |
| 11 | \( 1 + 3.07T + 11T^{2} \) |
| 13 | \( 1 - 5.67T + 13T^{2} \) |
| 17 | \( 1 - 7.16T + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 - 3.05T + 23T^{2} \) |
| 29 | \( 1 + 4.42T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 1.76T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 2.66T + 43T^{2} \) |
| 47 | \( 1 - 7.89T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 - 3.00T + 59T^{2} \) |
| 61 | \( 1 - 5.17T + 61T^{2} \) |
| 67 | \( 1 + 2.26T + 67T^{2} \) |
| 71 | \( 1 - 1.02T + 71T^{2} \) |
| 73 | \( 1 + 2.15T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 9.18T + 83T^{2} \) |
| 89 | \( 1 + 0.594T + 89T^{2} \) |
| 97 | \( 1 + 3.78T + 97T^{2} \) |
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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
−7.989975591426574735926117813892, −7.42085761937820916679755895399, −6.72772827305204016904025206839, −5.53463905009647335908501772423, −5.12259480591535576041086150504, −3.84356573993255959477076152980, −3.44713844830973293290433028038, −2.81668708489101330266961442339, −1.75228970592060642179040224213, −1.29988039210943843403916239757,
1.29988039210943843403916239757, 1.75228970592060642179040224213, 2.81668708489101330266961442339, 3.44713844830973293290433028038, 3.84356573993255959477076152980, 5.12259480591535576041086150504, 5.53463905009647335908501772423, 6.72772827305204016904025206839, 7.42085761937820916679755895399, 7.989975591426574735926117813892
