| L(s) = 1 | − 2·3-s + 5-s − 7-s + 9-s − 11-s + 4·13-s − 2·15-s − 4·19-s + 2·21-s + 25-s + 4·27-s + 6·29-s + 10·31-s + 2·33-s − 35-s − 2·37-s − 8·39-s − 12·41-s − 4·43-s + 45-s − 6·47-s + 49-s + 6·53-s − 55-s + 8·57-s − 6·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.516·15-s − 0.917·19-s + 0.436·21-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 1.79·31-s + 0.348·33-s − 0.169·35-s − 0.328·37-s − 1.28·39-s − 1.87·41-s − 0.609·43-s + 0.149·45-s − 0.875·47-s + 1/7·49-s + 0.824·53-s − 0.134·55-s + 1.05·57-s − 0.781·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.83726529079864, −15.23039991591595, −14.64454053103153, −13.86993478472691, −13.34821453626444, −13.14351172822373, −12.13359166180068, −11.99556911838553, −11.36359677146234, −10.62705098962537, −10.35264744609099, −9.928742512627051, −8.956684227388355, −8.490455745907191, −8.031221643149067, −6.885949748984600, −6.529694408506379, −6.142623792628039, −5.514859568044735, −4.851318391261979, −4.346374724940397, −3.352793711101806, −2.763835317794192, −1.738538006548195, −0.9345515157816413, 0,
0.9345515157816413, 1.738538006548195, 2.763835317794192, 3.352793711101806, 4.346374724940397, 4.851318391261979, 5.514859568044735, 6.142623792628039, 6.529694408506379, 6.885949748984600, 8.031221643149067, 8.490455745907191, 8.956684227388355, 9.928742512627051, 10.35264744609099, 10.62705098962537, 11.36359677146234, 11.99556911838553, 12.13359166180068, 13.14351172822373, 13.34821453626444, 13.86993478472691, 14.64454053103153, 15.23039991591595, 15.83726529079864
