L(s) = 1 | + 7.39·2-s − 9·3-s + 22.7·4-s − 66.5·6-s − 150.·7-s − 68.5·8-s + 81·9-s + 121·11-s − 204.·12-s + 868.·13-s − 1.11e3·14-s − 1.23e3·16-s + 2.31e3·17-s + 599.·18-s − 2.65e3·19-s + 1.35e3·21-s + 895.·22-s + 2.53e3·23-s + 616.·24-s + 6.42e3·26-s − 729·27-s − 3.43e3·28-s − 819.·29-s + 7.30e3·31-s − 6.94e3·32-s − 1.08e3·33-s + 1.71e4·34-s + ⋯ |
L(s) = 1 | + 1.30·2-s − 0.577·3-s + 0.710·4-s − 0.755·6-s − 1.16·7-s − 0.378·8-s + 0.333·9-s + 0.301·11-s − 0.410·12-s + 1.42·13-s − 1.52·14-s − 1.20·16-s + 1.94·17-s + 0.435·18-s − 1.68·19-s + 0.671·21-s + 0.394·22-s + 1.00·23-s + 0.218·24-s + 1.86·26-s − 0.192·27-s − 0.826·28-s − 0.181·29-s + 1.36·31-s − 1.19·32-s − 0.174·33-s + 2.54·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
good | 2 | \( 1 - 7.39T + 32T^{2} \) |
| 7 | \( 1 + 150.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 868.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.31e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.65e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.53e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 819.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.30e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.56e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.02e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.44e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.33e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.72e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.77e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.71e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.29e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.14e5T + 8.58e9T^{2} \) |
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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
−9.143988326429303449030078250055, −8.155065759041971791732497605002, −6.67020539443906456406104992186, −6.31545483956867836398483689529, −5.56706614436997597276697925830, −4.54022993360139095321513018454, −3.59557991808413660480953148371, −3.01034552446950475565957711200, −1.30744944848565061126168142289, 0,
1.30744944848565061126168142289, 3.01034552446950475565957711200, 3.59557991808413660480953148371, 4.54022993360139095321513018454, 5.56706614436997597276697925830, 6.31545483956867836398483689529, 6.67020539443906456406104992186, 8.155065759041971791732497605002, 9.143988326429303449030078250055