| L(s) = 1 | − 25·5-s − 56·7-s + 156·11-s + 350·13-s − 786·17-s − 740·19-s + 2.37e3·23-s + 625·25-s − 2.57e3·29-s + 4.57e3·31-s + 1.40e3·35-s − 1.22e4·37-s + 1.02e4·41-s + 1.60e4·43-s + 864·47-s − 1.36e4·49-s + 1.76e4·53-s − 3.90e3·55-s + 4.86e4·59-s − 3.37e4·61-s − 8.75e3·65-s − 3.52e3·67-s + 3.82e4·71-s − 7.97e4·73-s − 8.73e3·77-s − 9.92e4·79-s − 2.22e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.431·7-s + 0.388·11-s + 0.574·13-s − 0.659·17-s − 0.470·19-s + 0.936·23-s + 1/5·25-s − 0.568·29-s + 0.855·31-s + 0.193·35-s − 1.46·37-s + 0.950·41-s + 1.32·43-s + 0.0570·47-s − 0.813·49-s + 0.863·53-s − 0.173·55-s + 1.82·59-s − 1.16·61-s − 0.256·65-s − 0.0959·67-s + 0.901·71-s − 1.75·73-s − 0.167·77-s − 1.78·79-s − 0.355·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 + 8 p T + p^{5} T^{2} \) |
| 11 | \( 1 - 156 T + p^{5} T^{2} \) |
| 13 | \( 1 - 350 T + p^{5} T^{2} \) |
| 17 | \( 1 + 786 T + p^{5} T^{2} \) |
| 19 | \( 1 + 740 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2376 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2574 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4576 T + p^{5} T^{2} \) |
| 37 | \( 1 + 12202 T + p^{5} T^{2} \) |
| 41 | \( 1 - 10230 T + p^{5} T^{2} \) |
| 43 | \( 1 - 16084 T + p^{5} T^{2} \) |
| 47 | \( 1 - 864 T + p^{5} T^{2} \) |
| 53 | \( 1 - 17658 T + p^{5} T^{2} \) |
| 59 | \( 1 - 48684 T + p^{5} T^{2} \) |
| 61 | \( 1 + 33778 T + p^{5} T^{2} \) |
| 67 | \( 1 + 3524 T + p^{5} T^{2} \) |
| 71 | \( 1 - 38280 T + p^{5} T^{2} \) |
| 73 | \( 1 + 79702 T + p^{5} T^{2} \) |
| 79 | \( 1 + 99248 T + p^{5} T^{2} \) |
| 83 | \( 1 + 22284 T + p^{5} T^{2} \) |
| 89 | \( 1 + 94650 T + p^{5} T^{2} \) |
| 97 | \( 1 - 9122 T + p^{5} T^{2} \) |
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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
−9.046599606685395139230407510616, −8.543368190940214111893098002595, −7.37179366670562566353707507736, −6.64552316151574683603464620687, −5.71755341008125249451322399447, −4.50598548298926642627342844805, −3.68001160289541791245477210101, −2.58909486384138577016222396615, −1.20843969467041333308331664752, 0,
1.20843969467041333308331664752, 2.58909486384138577016222396615, 3.68001160289541791245477210101, 4.50598548298926642627342844805, 5.71755341008125249451322399447, 6.64552316151574683603464620687, 7.37179366670562566353707507736, 8.543368190940214111893098002595, 9.046599606685395139230407510616
