| L(s) = 1 | + 10·2-s + 68·4-s + 56·5-s − 49·7-s + 360·8-s + 560·10-s − 232·11-s − 140·13-s − 490·14-s + 1.42e3·16-s + 1.72e3·17-s − 98·19-s + 3.80e3·20-s − 2.32e3·22-s − 1.82e3·23-s + 11·25-s − 1.40e3·26-s − 3.33e3·28-s − 3.41e3·29-s − 7.64e3·31-s + 2.72e3·32-s + 1.72e4·34-s − 2.74e3·35-s − 1.03e4·37-s − 980·38-s + 2.01e4·40-s + 1.79e4·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 17/8·4-s + 1.00·5-s − 0.377·7-s + 1.98·8-s + 1.77·10-s − 0.578·11-s − 0.229·13-s − 0.668·14-s + 1.39·16-s + 1.44·17-s − 0.0622·19-s + 2.12·20-s − 1.02·22-s − 0.718·23-s + 0.00351·25-s − 0.406·26-s − 0.803·28-s − 0.754·29-s − 1.42·31-s + 0.469·32-s + 2.55·34-s − 0.378·35-s − 1.24·37-s − 0.110·38-s + 1.99·40-s + 1.66·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.975203293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.975203293\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 - 5 p T + p^{5} T^{2} \) |
| 5 | \( 1 - 56 T + p^{5} T^{2} \) |
| 11 | \( 1 + 232 T + p^{5} T^{2} \) |
| 13 | \( 1 + 140 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1722 T + p^{5} T^{2} \) |
| 19 | \( 1 + 98 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1824 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3418 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7644 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10398 T + p^{5} T^{2} \) |
| 41 | \( 1 - 17962 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10880 T + p^{5} T^{2} \) |
| 47 | \( 1 + 9324 T + p^{5} T^{2} \) |
| 53 | \( 1 + 2262 T + p^{5} T^{2} \) |
| 59 | \( 1 - 2730 T + p^{5} T^{2} \) |
| 61 | \( 1 - 25648 T + p^{5} T^{2} \) |
| 67 | \( 1 + 48404 T + p^{5} T^{2} \) |
| 71 | \( 1 - 58560 T + p^{5} T^{2} \) |
| 73 | \( 1 - 68082 T + p^{5} T^{2} \) |
| 79 | \( 1 - 31784 T + p^{5} T^{2} \) |
| 83 | \( 1 - 20538 T + p^{5} T^{2} \) |
| 89 | \( 1 - 50582 T + p^{5} T^{2} \) |
| 97 | \( 1 + 58506 T + p^{5} T^{2} \) |
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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
−13.94913286263688147168190464311, −12.96812867206438203949581105958, −12.25188367449539334752303065913, −10.83115965099407741695904204146, −9.639434028993440379376197880531, −7.46641612547275078022193965130, −6.01506616882935937543720922543, −5.29641825413961227450840317371, −3.59013564119637838204717865625, −2.14810415613733623647912509178,
2.14810415613733623647912509178, 3.59013564119637838204717865625, 5.29641825413961227450840317371, 6.01506616882935937543720922543, 7.46641612547275078022193965130, 9.639434028993440379376197880531, 10.83115965099407741695904204146, 12.25188367449539334752303065913, 12.96812867206438203949581105958, 13.94913286263688147168190464311
