| L(s) = 1 | − 2.30·2-s + 3.30·4-s + 5-s − 2.60·7-s − 3.00·8-s − 2.30·10-s + 4.60·11-s + 6.60·13-s + 6·14-s + 0.302·16-s + 1.60·17-s + 3.60·19-s + 3.30·20-s − 10.6·22-s − 3·23-s + 25-s − 15.2·26-s − 8.60·28-s + 1.39·29-s − 5.60·31-s + 5.30·32-s − 3.69·34-s − 2.60·35-s + 2·37-s − 8.30·38-s − 3.00·40-s − 4.60·41-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.65·4-s + 0.447·5-s − 0.984·7-s − 1.06·8-s − 0.728·10-s + 1.38·11-s + 1.83·13-s + 1.60·14-s + 0.0756·16-s + 0.389·17-s + 0.827·19-s + 0.738·20-s − 2.26·22-s − 0.625·23-s + 0.200·25-s − 2.98·26-s − 1.62·28-s + 0.258·29-s − 1.00·31-s + 0.937·32-s − 0.634·34-s − 0.440·35-s + 0.328·37-s − 1.34·38-s − 0.474·40-s − 0.719·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5751167207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5751167207\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 17 | \( 1 - 1.60T + 17T^{2} \) |
| 19 | \( 1 - 3.60T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 1.39T + 29T^{2} \) |
| 31 | \( 1 + 5.60T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.60T + 41T^{2} \) |
| 43 | \( 1 - 0.605T + 43T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 - 1.60T + 53T^{2} \) |
| 59 | \( 1 + 1.39T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 + 0.788T + 67T^{2} \) |
| 71 | \( 1 - 7.39T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−13.21313702094556918216664271616, −11.83778801607994380884494809492, −10.90185703185258061432307942083, −9.792658747522122841896446112431, −9.217039199644101934958400493644, −8.259470209742334016363486665276, −6.83689533944602881161552624859, −6.07410054170707884004360537213, −3.53742582758442874558829537550, −1.38607956939075027902575154310,
1.38607956939075027902575154310, 3.53742582758442874558829537550, 6.07410054170707884004360537213, 6.83689533944602881161552624859, 8.259470209742334016363486665276, 9.217039199644101934958400493644, 9.792658747522122841896446112431, 10.90185703185258061432307942083, 11.83778801607994380884494809492, 13.21313702094556918216664271616
