| L(s) = 1 | + 2.30·2-s + 3.30·4-s + 5-s + 3.00·8-s + 2.30·10-s − 4.60·11-s − 6.60·13-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 − 1.39·29-s + 5.60·31-s − 5.30·32-s + 3.69·34-s + 2·37-s − 8.30·38-s + 3.00·40-s − 4.60·41-s + 0.605·43-s − 15.2·44-s + 6.90·46-s − 9.21·47-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.65·4-s + 0.447·5-s + 1.06·8-s + 0.728·10-s − 1.38·11-s − 1.83·13-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 − 0.258·29-s + 1.00·31-s − 0.937·32-s + 0.634·34-s + 0.328·37-s − 1.34·38-s + 0.474·40-s − 0.719·41-s + 0.0923·43-s − 2.29·44-s + 1.01·46-s − 1.34·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 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
−7.37701363772174082345710457902, −6.78236040107220290859438709139, −5.99033498582537908823329134988, −5.28774891879322180321716117890, −4.86738034425495624301322104374, −4.26575043973260577168018833979, −3.00225923306642540567578889823, −2.71659540296893050365706968988, −1.85124376788983485860466004642, 0,
1.85124376788983485860466004642, 2.71659540296893050365706968988, 3.00225923306642540567578889823, 4.26575043973260577168018833979, 4.86738034425495624301322104374, 5.28774891879322180321716117890, 5.99033498582537908823329134988, 6.78236040107220290859438709139, 7.37701363772174082345710457902
