| L(s) = 1 | − 2·2-s + 3·3-s − 4·4-s − 6·6-s − 7·7-s + 24·8-s + 9·9-s − 21·11-s − 12·12-s + 24·13-s + 14·14-s − 16·16-s − 22·17-s − 18·18-s + 16·19-s − 21·21-s + 42·22-s − 25·23-s + 72·24-s − 48·26-s + 27·27-s + 28·28-s + 167·29-s + 10·31-s − 160·32-s − 63·33-s + 44·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.575·11-s − 0.288·12-s + 0.512·13-s + 0.267·14-s − 1/4·16-s − 0.313·17-s − 0.235·18-s + 0.193·19-s − 0.218·21-s + 0.407·22-s − 0.226·23-s + 0.612·24-s − 0.362·26-s + 0.192·27-s + 0.188·28-s + 1.06·29-s + 0.0579·31-s − 0.883·32-s − 0.332·33-s + 0.221·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 + p T + p^{3} T^{2} \) |
| 11 | \( 1 + 21 T + p^{3} T^{2} \) |
| 13 | \( 1 - 24 T + p^{3} T^{2} \) |
| 17 | \( 1 + 22 T + p^{3} T^{2} \) |
| 19 | \( 1 - 16 T + p^{3} T^{2} \) |
| 23 | \( 1 + 25 T + p^{3} T^{2} \) |
| 29 | \( 1 - 167 T + p^{3} T^{2} \) |
| 31 | \( 1 - 10 T + p^{3} T^{2} \) |
| 37 | \( 1 + 133 T + p^{3} T^{2} \) |
| 41 | \( 1 + 168 T + p^{3} T^{2} \) |
| 43 | \( 1 + 97 T + p^{3} T^{2} \) |
| 47 | \( 1 + 400 T + p^{3} T^{2} \) |
| 53 | \( 1 + 182 T + p^{3} T^{2} \) |
| 59 | \( 1 - 488 T + p^{3} T^{2} \) |
| 61 | \( 1 - 28 T + p^{3} T^{2} \) |
| 67 | \( 1 + 967 T + p^{3} T^{2} \) |
| 71 | \( 1 + 285 T + p^{3} T^{2} \) |
| 73 | \( 1 + 838 T + p^{3} T^{2} \) |
| 79 | \( 1 + 469 T + p^{3} T^{2} \) |
| 83 | \( 1 + 406 T + p^{3} T^{2} \) |
| 89 | \( 1 - 324 T + p^{3} T^{2} \) |
| 97 | \( 1 + 114 T + p^{3} T^{2} \) |
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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
−9.994363632441631496253306497444, −8.988913883312578577791866242614, −8.413238844581366872977410908542, −7.59445724470201683820816399428, −6.54721555867566781099203619860, −5.19156374624135818127915049649, −4.14071592042000735935794701959, −2.96972317589706725819097077895, −1.47864185808319947753407820245, 0,
1.47864185808319947753407820245, 2.96972317589706725819097077895, 4.14071592042000735935794701959, 5.19156374624135818127915049649, 6.54721555867566781099203619860, 7.59445724470201683820816399428, 8.413238844581366872977410908542, 8.988913883312578577791866242614, 9.994363632441631496253306497444
