| L(s) = 1 | − 2·2-s + 4·4-s + 12·5-s + 7·7-s − 8·8-s − 24·10-s − 48·11-s + 56·13-s − 14·14-s + 16·16-s + 114·17-s + 2·19-s + 48·20-s + 96·22-s + 120·23-s + 19·25-s − 112·26-s + 28·28-s + 54·29-s + 236·31-s − 32·32-s − 228·34-s + 84·35-s + 146·37-s − 4·38-s − 96·40-s − 126·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.07·5-s + 0.377·7-s − 0.353·8-s − 0.758·10-s − 1.31·11-s + 1.19·13-s − 0.267·14-s + 1/4·16-s + 1.62·17-s + 0.0241·19-s + 0.536·20-s + 0.930·22-s + 1.08·23-s + 0.151·25-s − 0.844·26-s + 0.188·28-s + 0.345·29-s + 1.36·31-s − 0.176·32-s − 1.15·34-s + 0.405·35-s + 0.648·37-s − 0.0170·38-s − 0.379·40-s − 0.479·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.477815432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.477815432\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 13 | \( 1 - 56 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 2 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 236 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 376 T + p^{3} T^{2} \) |
| 47 | \( 1 - 12 T + p^{3} T^{2} \) |
| 53 | \( 1 + 174 T + p^{3} T^{2} \) |
| 59 | \( 1 + 138 T + p^{3} T^{2} \) |
| 61 | \( 1 - 380 T + p^{3} T^{2} \) |
| 67 | \( 1 + 484 T + p^{3} T^{2} \) |
| 71 | \( 1 + 576 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1150 T + p^{3} T^{2} \) |
| 79 | \( 1 - 776 T + p^{3} T^{2} \) |
| 83 | \( 1 + 378 T + p^{3} T^{2} \) |
| 89 | \( 1 - 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1330 T + p^{3} T^{2} \) |
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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
−13.02570880484105993593698752747, −11.67423030315901212502035890382, −10.50617915547475220655387935512, −9.916888722053652703089617050958, −8.635401473848874599034373099109, −7.71352430583081297261582974056, −6.22374336310940453675116038868, −5.21725092469615003901875058970, −2.93186402518006212030117246276, −1.30489599730703285759753384598,
1.30489599730703285759753384598, 2.93186402518006212030117246276, 5.21725092469615003901875058970, 6.22374336310940453675116038868, 7.71352430583081297261582974056, 8.635401473848874599034373099109, 9.916888722053652703089617050958, 10.50617915547475220655387935512, 11.67423030315901212502035890382, 13.02570880484105993593698752747
