L(s) = 1 | − 3.09i·3-s + (−1.73 + 1.41i)5-s − 4.37·7-s − 6.58·9-s − 2.53i·11-s + (2.64 + 2.44i)13-s + (4.37 + 5.36i)15-s − 1.29i·17-s − 5.36i·19-s + 13.5i·21-s − 1.80i·23-s + (0.999 − 4.89i)25-s + 11.0i·27-s − 7.58·29-s − 3.12i·31-s + ⋯ |
L(s) = 1 | − 1.78i·3-s + (−0.774 + 0.632i)5-s − 1.65·7-s − 2.19·9-s − 0.763i·11-s + (0.733 + 0.679i)13-s + (1.13 + 1.38i)15-s − 0.313i·17-s − 1.23i·19-s + 2.95i·21-s − 0.376i·23-s + (0.199 − 0.979i)25-s + 2.13i·27-s − 1.40·29-s − 0.561i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0163845 + 0.526839i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0163845 + 0.526839i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.73 - 1.41i)T \) |
| 13 | \( 1 + (-2.64 - 2.44i)T \) |
good | 3 | \( 1 + 3.09iT - 3T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 + 2.53iT - 11T^{2} \) |
| 17 | \( 1 + 1.29iT - 17T^{2} \) |
| 19 | \( 1 + 5.36iT - 19T^{2} \) |
| 23 | \( 1 + 1.80iT - 23T^{2} \) |
| 29 | \( 1 + 7.58T + 29T^{2} \) |
| 31 | \( 1 + 3.12iT - 31T^{2} \) |
| 37 | \( 1 - 2.55T + 37T^{2} \) |
| 41 | \( 1 + 7.89iT - 41T^{2} \) |
| 43 | \( 1 - 4.38iT - 43T^{2} \) |
| 47 | \( 1 + 6.20T + 47T^{2} \) |
| 53 | \( 1 + 6.19iT - 53T^{2} \) |
| 59 | \( 1 - 10.4iT - 59T^{2} \) |
| 61 | \( 1 - 3.58T + 61T^{2} \) |
| 67 | \( 1 - 2.55T + 67T^{2} \) |
| 71 | \( 1 + 0.295iT - 71T^{2} \) |
| 73 | \( 1 - 2.55T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 0.723T + 83T^{2} \) |
| 89 | \( 1 - 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−11.59423578876109662603344123235, −10.98938691490538410353209477872, −9.359044592996471077765778858235, −8.392999372718258201157448446609, −7.24689414994526769088563743057, −6.68618542342886222824117726417, −5.95658148771736566744097880556, −3.60092490133216161012012192919, −2.57360796458670893059355337453, −0.39571677344501970876545734483,
3.40129154430712144561358409556, 3.85557078244062818850473660291, 5.14106146244585628738676197479, 6.18677614397187601616851277512, 7.84914880551161585435326343201, 8.960494343122640108930978623020, 9.686927294932541532772178224144, 10.33250807743382126823633119259, 11.34319304880590229992783548355, 12.46298434435879933136255968680