| L(s) = 1 | + 2-s − 0.237·3-s + 4-s + 4.10·5-s − 0.237·6-s + 4.01·7-s + 8-s − 2.94·9-s + 4.10·10-s + 5.87·11-s − 0.237·12-s + 2.21·13-s + 4.01·14-s − 0.975·15-s + 16-s − 6.80·17-s − 2.94·18-s − 5.35·19-s + 4.10·20-s − 0.953·21-s + 5.87·22-s − 0.252·23-s − 0.237·24-s + 11.8·25-s + 2.21·26-s + 1.41·27-s + 4.01·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.137·3-s + 0.5·4-s + 1.83·5-s − 0.0969·6-s + 1.51·7-s + 0.353·8-s − 0.981·9-s + 1.29·10-s + 1.77·11-s − 0.0685·12-s + 0.614·13-s + 1.07·14-s − 0.251·15-s + 0.250·16-s − 1.65·17-s − 0.693·18-s − 1.22·19-s + 0.918·20-s − 0.208·21-s + 1.25·22-s − 0.0527·23-s − 0.0484·24-s + 2.37·25-s + 0.434·26-s + 0.271·27-s + 0.759·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.501366545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.501366545\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 0.237T + 3T^{2} \) |
| 5 | \( 1 - 4.10T + 5T^{2} \) |
| 7 | \( 1 - 4.01T + 7T^{2} \) |
| 11 | \( 1 - 5.87T + 11T^{2} \) |
| 13 | \( 1 - 2.21T + 13T^{2} \) |
| 17 | \( 1 + 6.80T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 + 0.252T + 23T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 9.37T + 41T^{2} \) |
| 43 | \( 1 + 5.72T + 43T^{2} \) |
| 47 | \( 1 + 3.19T + 47T^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 - 9.54T + 61T^{2} \) |
| 67 | \( 1 + 0.0925T + 67T^{2} \) |
| 71 | \( 1 + 2.16T + 71T^{2} \) |
| 73 | \( 1 + 0.264T + 73T^{2} \) |
| 79 | \( 1 - 8.52T + 79T^{2} \) |
| 83 | \( 1 + 9.53T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
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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
−8.466418440956661473309405468765, −6.95483919497467390421286252427, −6.34929898846733575265419881610, −6.07415154204754491376509319846, −5.18347729527176699141595747998, −4.60905915280412557586676033908, −3.83506459154006809387211762219, −2.51149205743069764234165642626, −1.94860145752911834206157640432, −1.27480440981728328295114727093,
1.27480440981728328295114727093, 1.94860145752911834206157640432, 2.51149205743069764234165642626, 3.83506459154006809387211762219, 4.60905915280412557586676033908, 5.18347729527176699141595747998, 6.07415154204754491376509319846, 6.34929898846733575265419881610, 6.95483919497467390421286252427, 8.466418440956661473309405468765
