| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 106·5-s + 36·6-s − 49·7-s − 64·8-s + 81·9-s − 424·10-s + 121·11-s − 144·12-s − 166·13-s + 196·14-s − 954·15-s + 256·16-s − 842·17-s − 324·18-s + 1.20e3·19-s + 1.69e3·20-s + 441·21-s − 484·22-s + 1.72e3·23-s + 576·24-s + 8.11e3·25-s + 664·26-s − 729·27-s − 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.89·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.34·10-s + 0.301·11-s − 0.288·12-s − 0.272·13-s + 0.267·14-s − 1.09·15-s + 1/4·16-s − 0.706·17-s − 0.235·18-s + 0.762·19-s + 0.948·20-s + 0.218·21-s − 0.213·22-s + 0.679·23-s + 0.204·24-s + 2.59·25-s + 0.192·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.834456689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.834456689\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| 11 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 106 T + p^{5} T^{2} \) |
| 13 | \( 1 + 166 T + p^{5} T^{2} \) |
| 17 | \( 1 + 842 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1200 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1724 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1010 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1328 T + p^{5} T^{2} \) |
| 37 | \( 1 - 15718 T + p^{5} T^{2} \) |
| 41 | \( 1 + 7498 T + p^{5} T^{2} \) |
| 43 | \( 1 + 2456 T + p^{5} T^{2} \) |
| 47 | \( 1 + 2252 T + p^{5} T^{2} \) |
| 53 | \( 1 - 15714 T + p^{5} T^{2} \) |
| 59 | \( 1 + 5340 T + p^{5} T^{2} \) |
| 61 | \( 1 + 25718 T + p^{5} T^{2} \) |
| 67 | \( 1 + 4572 T + p^{5} T^{2} \) |
| 71 | \( 1 + 67708 T + p^{5} T^{2} \) |
| 73 | \( 1 + 7406 T + p^{5} T^{2} \) |
| 79 | \( 1 - 25560 T + p^{5} T^{2} \) |
| 83 | \( 1 - 22404 T + p^{5} T^{2} \) |
| 89 | \( 1 - 61530 T + p^{5} T^{2} \) |
| 97 | \( 1 + 72782 T + p^{5} T^{2} \) |
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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
−10.09102743765225684852442695398, −9.479652877596350909114979259516, −8.831182864765191209624874811535, −7.31041237773815890363399414506, −6.44114910603337716097141368927, −5.82497698713724101204712929717, −4.79820564255159375786425783097, −2.91215828110760874266415929364, −1.85066267877898136957015380714, −0.819340691045988662157109012381,
0.819340691045988662157109012381, 1.85066267877898136957015380714, 2.91215828110760874266415929364, 4.79820564255159375786425783097, 5.82497698713724101204712929717, 6.44114910603337716097141368927, 7.31041237773815890363399414506, 8.831182864765191209624874811535, 9.479652877596350909114979259516, 10.09102743765225684852442695398
