L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s − 0.999·8-s + (0.866 + 0.5i)9-s + (−0.866 − 0.499i)10-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)16-s + (−0.5 − 1.86i)17-s + 0.999i·18-s − 0.999i·20-s + (0.499 − 0.866i)25-s + (0.866 + 0.499i)26-s + (−0.866 + 0.5i)29-s + (0.499 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s − 0.999·8-s + (0.866 + 0.5i)9-s + (−0.866 − 0.499i)10-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)16-s + (−0.5 − 1.86i)17-s + 0.999i·18-s − 0.999i·20-s + (0.499 − 0.866i)25-s + (0.866 + 0.499i)26-s + (−0.866 + 0.5i)29-s + (0.499 − 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00418 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00418 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8267653783\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8267653783\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−12.61337232463396429039470225862, −11.62514505272491351275930965637, −10.73657818198926299008783825362, −9.386871196467819744932731786614, −8.261245724934543746971149499397, −7.36866324412574067608422162130, −6.72944299528720440298427827168, −5.26284332385209644738563352145, −4.23228271926912416754467420799, −3.05971864672212594542885710970,
1.62203988506869908403098881593, 3.77569788738540120686663690816, 4.17384474787399573304638830110, 5.68704839423958358998488005224, 6.89070881870377856157680103898, 8.391931884171576661919441095690, 9.149893248015773131784832409971, 10.35372006464948839885250191313, 11.14612006845730062092668287834, 12.06923163799886955169739283318