| L(s) = 1 | − 4·2-s + 16·4-s + 81·5-s − 58·7-s − 64·8-s − 324·10-s + 300·11-s − 16·13-s + 232·14-s + 256·16-s − 222·17-s − 1.90e3·19-s + 1.29e3·20-s − 1.20e3·22-s − 1.60e3·23-s + 3.43e3·25-s + 64·26-s − 928·28-s − 2.37e3·29-s − 3.55e3·31-s − 1.02e3·32-s + 888·34-s − 4.69e3·35-s − 9.91e3·37-s + 7.61e3·38-s − 5.18e3·40-s − 9.77e3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.44·5-s − 0.447·7-s − 0.353·8-s − 1.02·10-s + 0.747·11-s − 0.0262·13-s + 0.316·14-s + 1/4·16-s − 0.186·17-s − 1.20·19-s + 0.724·20-s − 0.528·22-s − 0.632·23-s + 1.09·25-s + 0.0185·26-s − 0.223·28-s − 0.523·29-s − 0.663·31-s − 0.176·32-s + 0.131·34-s − 0.648·35-s − 1.19·37-s + 0.855·38-s − 0.512·40-s − 0.908·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 81 T + p^{5} T^{2} \) |
| 7 | \( 1 + 58 T + p^{5} T^{2} \) |
| 11 | \( 1 - 300 T + p^{5} T^{2} \) |
| 13 | \( 1 + 16 T + p^{5} T^{2} \) |
| 17 | \( 1 + 222 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1903 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1605 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2373 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3550 T + p^{5} T^{2} \) |
| 37 | \( 1 + 268 p T + p^{5} T^{2} \) |
| 41 | \( 1 + 9774 T + p^{5} T^{2} \) |
| 43 | \( 1 - 17876 T + p^{5} T^{2} \) |
| 47 | \( 1 + 23439 T + p^{5} T^{2} \) |
| 53 | \( 1 + 18489 T + p^{5} T^{2} \) |
| 59 | \( 1 - 19356 T + p^{5} T^{2} \) |
| 61 | \( 1 + 20902 T + p^{5} T^{2} \) |
| 67 | \( 1 - 48437 T + p^{5} T^{2} \) |
| 71 | \( 1 - 15465 T + p^{5} T^{2} \) |
| 73 | \( 1 - 84017 T + p^{5} T^{2} \) |
| 79 | \( 1 - 90080 T + p^{5} T^{2} \) |
| 83 | \( 1 + 104574 T + p^{5} T^{2} \) |
| 89 | \( 1 - 46020 T + p^{5} T^{2} \) |
| 97 | \( 1 - 73583 T + p^{5} T^{2} \) |
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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
−9.608727492868307150333137122237, −9.112159461625729039461047409237, −8.112209039345582031737177616232, −6.71450745824809083795442645333, −6.30533537706078353856531459466, −5.23100278739501974952985442507, −3.69681938205087118188699401970, −2.28263878449479121771075700529, −1.53559827065362487201359296670, 0,
1.53559827065362487201359296670, 2.28263878449479121771075700529, 3.69681938205087118188699401970, 5.23100278739501974952985442507, 6.30533537706078353856531459466, 6.71450745824809083795442645333, 8.112209039345582031737177616232, 9.112159461625729039461047409237, 9.608727492868307150333137122237
