| L(s) = 1 | + 1.70·3-s − 5-s − 0.630·7-s − 0.0783·9-s − 3.70·11-s − 3.07·13-s − 1.70·15-s − 6.04·17-s − 1.36·19-s − 1.07·21-s + 1.70·23-s + 25-s − 5.26·27-s + 29-s − 4.78·31-s − 6.34·33-s + 0.630·35-s − 2.63·37-s − 5.26·39-s + 8.34·41-s + 1.12·43-s + 0.0783·45-s + 10.3·47-s − 6.60·49-s − 10.3·51-s + 9.02·53-s + 3.70·55-s + ⋯ |
| L(s) = 1 | + 0.986·3-s − 0.447·5-s − 0.238·7-s − 0.0261·9-s − 1.11·11-s − 0.853·13-s − 0.441·15-s − 1.46·17-s − 0.314·19-s − 0.235·21-s + 0.356·23-s + 0.200·25-s − 1.01·27-s + 0.185·29-s − 0.859·31-s − 1.10·33-s + 0.106·35-s − 0.432·37-s − 0.842·39-s + 1.30·41-s + 0.171·43-s + 0.0116·45-s + 1.51·47-s − 0.943·49-s − 1.44·51-s + 1.23·53-s + 0.500·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.70T + 3T^{2} \) |
| 7 | \( 1 + 0.630T + 7T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 + 3.07T + 13T^{2} \) |
| 17 | \( 1 + 6.04T + 17T^{2} \) |
| 19 | \( 1 + 1.36T + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 + 2.63T + 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 9.02T + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 + 9.60T + 61T^{2} \) |
| 67 | \( 1 + 8.44T + 67T^{2} \) |
| 71 | \( 1 + 9.75T + 71T^{2} \) |
| 73 | \( 1 + 4.29T + 73T^{2} \) |
| 79 | \( 1 - 6.23T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−9.051152687093268761748124520537, −8.752577536541371190131994321339, −7.66969100124969520993081286884, −7.25699716096968861738755498117, −6.01581734009807422060920750782, −4.93240679859869155194793535534, −4.00752715486541866783501543151, −2.86580362066059098182257105810, −2.24726801042318342357932078609, 0,
2.24726801042318342357932078609, 2.86580362066059098182257105810, 4.00752715486541866783501543151, 4.93240679859869155194793535534, 6.01581734009807422060920750782, 7.25699716096968861738755498117, 7.66969100124969520993081286884, 8.752577536541371190131994321339, 9.051152687093268761748124520537
