| L(s) = 1 | + 1.50·2-s + 3-s + 0.272·4-s + 5-s + 1.50·6-s − 2.60·8-s + 9-s + 1.50·10-s + 11-s + 0.272·12-s + 1.59·13-s + 15-s − 4.47·16-s + 1.65·17-s + 1.50·18-s − 5.58·19-s + 0.272·20-s + 1.50·22-s + 1.01·23-s − 2.60·24-s + 25-s + 2.40·26-s + 27-s − 0.184·29-s + 1.50·30-s − 4.97·31-s − 1.52·32-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.577·3-s + 0.136·4-s + 0.447·5-s + 0.615·6-s − 0.920·8-s + 0.333·9-s + 0.476·10-s + 0.301·11-s + 0.0785·12-s + 0.442·13-s + 0.258·15-s − 1.11·16-s + 0.402·17-s + 0.355·18-s − 1.28·19-s + 0.0608·20-s + 0.321·22-s + 0.212·23-s − 0.531·24-s + 0.200·25-s + 0.471·26-s + 0.192·27-s − 0.0341·29-s + 0.275·30-s − 0.893·31-s − 0.270·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.458711977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.458711977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.50T + 2T^{2} \) |
| 13 | \( 1 - 1.59T + 13T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 - 1.01T + 23T^{2} \) |
| 29 | \( 1 + 0.184T + 29T^{2} \) |
| 31 | \( 1 + 4.97T + 31T^{2} \) |
| 37 | \( 1 - 3.25T + 37T^{2} \) |
| 41 | \( 1 - 9.95T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 1.62T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 5.19T + 71T^{2} \) |
| 73 | \( 1 - 2.30T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.79418893882220910657630639930, −6.98320781898605489397397341456, −6.19854313277367412913317361155, −5.75504777453908705173716686289, −4.96241802080049450314606300636, −4.04229085459938034272296935289, −3.82997320005444935377290438373, −2.72217466305283959598129084000, −2.17855101833852427941916870856, −0.869028283563726498208853410979,
0.869028283563726498208853410979, 2.17855101833852427941916870856, 2.72217466305283959598129084000, 3.82997320005444935377290438373, 4.04229085459938034272296935289, 4.96241802080049450314606300636, 5.75504777453908705173716686289, 6.19854313277367412913317361155, 6.98320781898605489397397341456, 7.79418893882220910657630639930
