| L(s) = 1 | − 16·2-s − 174·3-s + 256·4-s + 2.78e3·6-s − 4.65e3·7-s − 4.09e3·8-s + 1.05e4·9-s + 2.89e4·11-s − 4.45e4·12-s + 1.64e5·13-s + 7.45e4·14-s + 6.55e4·16-s + 5.94e5·17-s − 1.69e5·18-s − 2.95e5·19-s + 8.10e5·21-s − 4.63e5·22-s − 2.54e6·23-s + 7.12e5·24-s − 2.63e6·26-s + 1.58e6·27-s − 1.19e6·28-s − 3.72e6·29-s + 2.33e6·31-s − 1.04e6·32-s − 5.04e6·33-s − 9.51e6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.24·3-s + 1/2·4-s + 0.876·6-s − 0.733·7-s − 0.353·8-s + 0.538·9-s + 0.597·11-s − 0.620·12-s + 1.59·13-s + 0.518·14-s + 1/4·16-s + 1.72·17-s − 0.380·18-s − 0.520·19-s + 0.909·21-s − 0.422·22-s − 1.89·23-s + 0.438·24-s − 1.12·26-s + 0.572·27-s − 0.366·28-s − 0.977·29-s + 0.454·31-s − 0.176·32-s − 0.740·33-s − 1.22·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{4} T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 58 p T + p^{9} T^{2} \) |
| 7 | \( 1 + 4658 T + p^{9} T^{2} \) |
| 11 | \( 1 - 28992 T + p^{9} T^{2} \) |
| 13 | \( 1 - 164446 T + p^{9} T^{2} \) |
| 17 | \( 1 - 594822 T + p^{9} T^{2} \) |
| 19 | \( 1 + 295780 T + p^{9} T^{2} \) |
| 23 | \( 1 + 2544534 T + p^{9} T^{2} \) |
| 29 | \( 1 + 3722970 T + p^{9} T^{2} \) |
| 31 | \( 1 - 2335772 T + p^{9} T^{2} \) |
| 37 | \( 1 + 10840418 T + p^{9} T^{2} \) |
| 41 | \( 1 - 21593862 T + p^{9} T^{2} \) |
| 43 | \( 1 + 10832294 T + p^{9} T^{2} \) |
| 47 | \( 1 + 5172138 T + p^{9} T^{2} \) |
| 53 | \( 1 + 98179674 T + p^{9} T^{2} \) |
| 59 | \( 1 - 16162860 T + p^{9} T^{2} \) |
| 61 | \( 1 + 43928158 T + p^{9} T^{2} \) |
| 67 | \( 1 - 81557422 T + p^{9} T^{2} \) |
| 71 | \( 1 - 161307732 T + p^{9} T^{2} \) |
| 73 | \( 1 - 247147966 T + p^{9} T^{2} \) |
| 79 | \( 1 + 583345720 T + p^{9} T^{2} \) |
| 83 | \( 1 - 14571786 T + p^{9} T^{2} \) |
| 89 | \( 1 - 470133690 T + p^{9} T^{2} \) |
| 97 | \( 1 - 117838462 T + p^{9} T^{2} \) |
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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
−12.60391152470825789708009252002, −11.68452692352710228192575360523, −10.66830224465832120892546287888, −9.620140852151869747197180421536, −8.129366860083110465068768947707, −6.45667777116602178484452557956, −5.78647623512196733467765745243, −3.63781577927730377934983938907, −1.31303774282062733519612420629, 0,
1.31303774282062733519612420629, 3.63781577927730377934983938907, 5.78647623512196733467765745243, 6.45667777116602178484452557956, 8.129366860083110465068768947707, 9.620140852151869747197180421536, 10.66830224465832120892546287888, 11.68452692352710228192575360523, 12.60391152470825789708009252002
