| L(s) = 1 | − 1.94·5-s − 0.969·7-s − 3·9-s + 0.202·11-s − 5.56·13-s + 3.01·17-s + 19-s + 6.09·23-s − 1.21·25-s − 6.46·29-s − 5.83·31-s + 1.88·35-s + 8.65·37-s − 7.53·41-s + 2.75·43-s + 5.83·45-s − 5.67·47-s − 6.06·49-s − 12.1·53-s − 0.393·55-s − 10.5·59-s + 1.88·61-s + 2.90·63-s + 10.8·65-s + 8.95·67-s + 5.27·71-s − 9.05·73-s + ⋯ |
| L(s) = 1 | − 0.869·5-s − 0.366·7-s − 9-s + 0.0609·11-s − 1.54·13-s + 0.730·17-s + 0.229·19-s + 1.27·23-s − 0.243·25-s − 1.20·29-s − 1.04·31-s + 0.318·35-s + 1.42·37-s − 1.17·41-s + 0.420·43-s + 0.869·45-s − 0.828·47-s − 0.865·49-s − 1.67·53-s − 0.0530·55-s − 1.37·59-s + 0.241·61-s + 0.366·63-s + 1.34·65-s + 1.09·67-s + 0.626·71-s − 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6835025011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6835025011\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 1.94T + 5T^{2} \) |
| 7 | \( 1 + 0.969T + 7T^{2} \) |
| 11 | \( 1 - 0.202T + 11T^{2} \) |
| 13 | \( 1 + 5.56T + 13T^{2} \) |
| 17 | \( 1 - 3.01T + 17T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 + 6.46T + 29T^{2} \) |
| 31 | \( 1 + 5.83T + 31T^{2} \) |
| 37 | \( 1 - 8.65T + 37T^{2} \) |
| 41 | \( 1 + 7.53T + 41T^{2} \) |
| 43 | \( 1 - 2.75T + 43T^{2} \) |
| 47 | \( 1 + 5.67T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 1.88T + 61T^{2} \) |
| 67 | \( 1 - 8.95T + 67T^{2} \) |
| 71 | \( 1 - 5.27T + 71T^{2} \) |
| 73 | \( 1 + 9.05T + 73T^{2} \) |
| 83 | \( 1 - 4.23T + 83T^{2} \) |
| 89 | \( 1 + 7.91T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.84141604147809409879900609986, −7.57129409426107967510583673259, −6.78342029995345896432091876983, −5.87979304161633866089469136725, −5.17396708019918584802637470266, −4.52546608637103164519521465513, −3.38300390972777565606983060855, −3.06703124627373126351930895887, −1.92189871611579277591957116742, −0.41258655127817816240804506498,
0.41258655127817816240804506498, 1.92189871611579277591957116742, 3.06703124627373126351930895887, 3.38300390972777565606983060855, 4.52546608637103164519521465513, 5.17396708019918584802637470266, 5.87979304161633866089469136725, 6.78342029995345896432091876983, 7.57129409426107967510583673259, 7.84141604147809409879900609986
