| L(s) = 1 | − 0.814·2-s − 1.33·4-s + 5-s + 2.87·7-s + 2.71·8-s − 0.814·10-s + 11-s + 0.395·13-s − 2.34·14-s + 0.458·16-s − 2.38·17-s − 19-s − 1.33·20-s − 0.814·22-s − 5.25·23-s + 25-s − 0.321·26-s − 3.84·28-s + 10.4·29-s + 2.07·31-s − 5.80·32-s + 1.94·34-s + 2.87·35-s − 10.0·37-s + 0.814·38-s + 2.71·40-s + 9.62·41-s + ⋯ |
| L(s) = 1 | − 0.576·2-s − 0.668·4-s + 0.447·5-s + 1.08·7-s + 0.960·8-s − 0.257·10-s + 0.301·11-s + 0.109·13-s − 0.626·14-s + 0.114·16-s − 0.577·17-s − 0.229·19-s − 0.298·20-s − 0.173·22-s − 1.09·23-s + 0.200·25-s − 0.0631·26-s − 0.726·28-s + 1.93·29-s + 0.371·31-s − 1.02·32-s + 0.332·34-s + 0.486·35-s − 1.65·37-s + 0.132·38-s + 0.429·40-s + 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.814T + 2T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 13 | \( 1 - 0.395T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 23 | \( 1 + 5.25T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 2.07T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 9.62T + 41T^{2} \) |
| 43 | \( 1 + 8.14T + 43T^{2} \) |
| 47 | \( 1 + 4.17T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 9.96T + 59T^{2} \) |
| 61 | \( 1 + 3.18T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 9.50T + 73T^{2} \) |
| 79 | \( 1 - 8.73T + 79T^{2} \) |
| 83 | \( 1 + 2.37T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 7.04T + 97T^{2} \) |
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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
−7.65226744812446421880330283943, −6.66131928641310764254669761601, −6.10178357964268850477633133847, −5.07329542985843155907704029427, −4.68153218281878328509437809011, −4.03331021712401567193174758232, −2.94374583794797384767358578426, −1.81798186979262811800834887443, −1.30812879160181593139783835375, 0,
1.30812879160181593139783835375, 1.81798186979262811800834887443, 2.94374583794797384767358578426, 4.03331021712401567193174758232, 4.68153218281878328509437809011, 5.07329542985843155907704029427, 6.10178357964268850477633133847, 6.66131928641310764254669761601, 7.65226744812446421880330283943
