| L(s) = 1 | + 0.765·3-s + 2.93·5-s + 1.41·7-s − 2.41·9-s − 4.46·11-s + 0.765·13-s + 2.24·15-s − 2.82·17-s − 4.01·19-s + 1.08·21-s − 8.24·23-s + 3.58·25-s − 4.14·27-s − 3.37·29-s + 4·31-s − 3.41·33-s + 4.14·35-s − 0.765·37-s + 0.585·39-s + 0.242·41-s − 5.09·43-s − 7.07·45-s − 0.343·47-s − 5·49-s − 2.16·51-s − 1.21·53-s − 13.0·55-s + ⋯ |
| L(s) = 1 | + 0.441·3-s + 1.31·5-s + 0.534·7-s − 0.804·9-s − 1.34·11-s + 0.212·13-s + 0.579·15-s − 0.685·17-s − 0.920·19-s + 0.236·21-s − 1.71·23-s + 0.717·25-s − 0.797·27-s − 0.627·29-s + 0.718·31-s − 0.594·33-s + 0.700·35-s − 0.125·37-s + 0.0938·39-s + 0.0378·41-s − 0.776·43-s − 1.05·45-s − 0.0500·47-s − 0.714·49-s − 0.303·51-s − 0.166·53-s − 1.76·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 0.765T + 3T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 4.46T + 11T^{2} \) |
| 13 | \( 1 - 0.765T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 4.01T + 19T^{2} \) |
| 23 | \( 1 + 8.24T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 0.765T + 37T^{2} \) |
| 41 | \( 1 - 0.242T + 41T^{2} \) |
| 43 | \( 1 + 5.09T + 43T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 + 1.21T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 + 1.84T + 61T^{2} \) |
| 67 | \( 1 - 5.99T + 67T^{2} \) |
| 71 | \( 1 + 8.24T + 71T^{2} \) |
| 73 | \( 1 - 9.89T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 4.90T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−8.313630025685768863634528520975, −7.49321939426648692084187693122, −6.34319939907105477512475167135, −5.89434665074686214914155942912, −5.17911307927478651227543517645, −4.35445396800044677249250071371, −3.17176982869938136850114005872, −2.24972520209923187294074008579, −1.88099880511095931526334869756, 0,
1.88099880511095931526334869756, 2.24972520209923187294074008579, 3.17176982869938136850114005872, 4.35445396800044677249250071371, 5.17911307927478651227543517645, 5.89434665074686214914155942912, 6.34319939907105477512475167135, 7.49321939426648692084187693122, 8.313630025685768863634528520975
