L(s) = 1 | + (−1.60 − 0.169i)2-s + (1.58 + 0.336i)4-s + (0.994 − 0.104i)5-s + (0.809 + 1.40i)7-s + (−0.951 − 0.309i)8-s − 1.61·10-s + (−0.994 − 0.104i)11-s + (−1.06 − 2.39i)14-s + (0.951 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (1.60 + 0.169i)20-s + (1.58 + 0.336i)22-s + (0.406 + 0.913i)23-s + (0.978 − 0.207i)25-s + (0.809 + 2.48i)28-s + (−0.459 − 0.413i)29-s + ⋯ |
L(s) = 1 | + (−1.60 − 0.169i)2-s + (1.58 + 0.336i)4-s + (0.994 − 0.104i)5-s + (0.809 + 1.40i)7-s + (−0.951 − 0.309i)8-s − 1.61·10-s + (−0.994 − 0.104i)11-s + (−1.06 − 2.39i)14-s + (0.951 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (1.60 + 0.169i)20-s + (1.58 + 0.336i)22-s + (0.406 + 0.913i)23-s + (0.978 − 0.207i)25-s + (0.809 + 2.48i)28-s + (−0.459 − 0.413i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6655913713\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6655913713\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.994 + 0.104i)T \) |
good | 2 | \( 1 + (1.60 + 0.169i)T + (0.978 + 0.207i)T^{2} \) |
| 7 | \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.994 + 0.104i)T + (0.978 + 0.207i)T^{2} \) |
| 13 | \( 1 + (-0.978 + 0.207i)T^{2} \) |
| 17 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.406 - 0.913i)T + (-0.669 + 0.743i)T^{2} \) |
| 29 | \( 1 + (0.459 + 0.413i)T + (0.104 + 0.994i)T^{2} \) |
| 31 | \( 1 + (-0.104 + 0.994i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (0.994 - 0.104i)T + (0.978 - 0.207i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.459 - 0.413i)T + (0.104 + 0.994i)T^{2} \) |
| 53 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 61 | \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
| 67 | \( 1 + (1.08 + 1.20i)T + (-0.104 + 0.994i)T^{2} \) |
| 71 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.413 - 0.459i)T + (-0.104 - 0.994i)T^{2} \) |
| 83 | \( 1 + (-0.336 - 1.58i)T + (-0.913 + 0.406i)T^{2} \) |
| 89 | \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \) |
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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
−9.423484722220268966347829207309, −8.758189752992025693713511626794, −8.028195089022224065604058132620, −7.63431281448436112547891472955, −6.28916093615378962212520272344, −5.62261181808532735778262774486, −4.93627177559431721313640695200, −3.07542198747582581182275132850, −2.12103829880132840031529713477, −1.51236131870347942395225268345,
0.861110488983252114048255101583, 1.86423148680276174403585449787, 2.93557197981825751611825195020, 4.52105560304018028912412213731, 5.28873624466954495649023696394, 6.48799695981347312557962895563, 7.17170650626240991410836869070, 7.72053122535061740126533931175, 8.472301573506011191934382791032, 9.229876693606574112154600664719