| L(s) = 1 | + 3·3-s − 12·5-s − 2·7-s + 9·9-s + 36·11-s + 13·13-s − 36·15-s − 78·17-s − 74·19-s − 6·21-s + 96·23-s + 19·25-s + 27·27-s + 18·29-s + 214·31-s + 108·33-s + 24·35-s − 286·37-s + 39·39-s − 384·41-s − 524·43-s − 108·45-s − 300·47-s − 339·49-s − 234·51-s + 558·53-s − 432·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.07·5-s − 0.107·7-s + 1/3·9-s + 0.986·11-s + 0.277·13-s − 0.619·15-s − 1.11·17-s − 0.893·19-s − 0.0623·21-s + 0.870·23-s + 0.151·25-s + 0.192·27-s + 0.115·29-s + 1.23·31-s + 0.569·33-s + 0.115·35-s − 1.27·37-s + 0.160·39-s − 1.46·41-s − 1.85·43-s − 0.357·45-s − 0.931·47-s − 0.988·49-s − 0.642·51-s + 1.44·53-s − 1.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 74 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 18 T + p^{3} T^{2} \) |
| 31 | \( 1 - 214 T + p^{3} T^{2} \) |
| 37 | \( 1 + 286 T + p^{3} T^{2} \) |
| 41 | \( 1 + 384 T + p^{3} T^{2} \) |
| 43 | \( 1 + 524 T + p^{3} T^{2} \) |
| 47 | \( 1 + 300 T + p^{3} T^{2} \) |
| 53 | \( 1 - 558 T + p^{3} T^{2} \) |
| 59 | \( 1 + 576 T + p^{3} T^{2} \) |
| 61 | \( 1 - 74 T + p^{3} T^{2} \) |
| 67 | \( 1 + 38 T + p^{3} T^{2} \) |
| 71 | \( 1 - 456 T + p^{3} T^{2} \) |
| 73 | \( 1 + 682 T + p^{3} T^{2} \) |
| 79 | \( 1 + 704 T + p^{3} T^{2} \) |
| 83 | \( 1 - 888 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 97 | \( 1 - 110 T + p^{3} T^{2} \) |
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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
−9.677338101599442514199099245115, −8.577309445388105919221761577971, −8.356025467404289236750310161098, −6.99420287962858719427672805937, −6.52459866292086647747060157393, −4.83003949684953888794828329645, −3.99528531785299648351322581938, −3.12659185447439819581869711337, −1.63540125795957993522501919198, 0,
1.63540125795957993522501919198, 3.12659185447439819581869711337, 3.99528531785299648351322581938, 4.83003949684953888794828329645, 6.52459866292086647747060157393, 6.99420287962858719427672805937, 8.356025467404289236750310161098, 8.577309445388105919221761577971, 9.677338101599442514199099245115
