| L(s) = 1 | − 0.212·2-s − 2.83·3-s − 1.95·4-s − 1.65·5-s + 0.602·6-s + 2.53·7-s + 0.839·8-s + 5.06·9-s + 0.352·10-s + 1.11·11-s + 5.55·12-s + 2.39·13-s − 0.539·14-s + 4.71·15-s + 3.73·16-s − 17-s − 1.07·18-s − 5.01·19-s + 3.24·20-s − 7.21·21-s − 0.236·22-s + 3.24·23-s − 2.38·24-s − 2.24·25-s − 0.508·26-s − 5.85·27-s − 4.96·28-s + ⋯ |
| L(s) = 1 | − 0.150·2-s − 1.63·3-s − 0.977·4-s − 0.741·5-s + 0.246·6-s + 0.959·7-s + 0.296·8-s + 1.68·9-s + 0.111·10-s + 0.336·11-s + 1.60·12-s + 0.664·13-s − 0.144·14-s + 1.21·15-s + 0.932·16-s − 0.242·17-s − 0.253·18-s − 1.15·19-s + 0.725·20-s − 1.57·21-s − 0.0504·22-s + 0.677·23-s − 0.486·24-s − 0.449·25-s − 0.0997·26-s − 1.12·27-s − 0.938·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 0.212T + 2T^{2} \) |
| 3 | \( 1 + 2.83T + 3T^{2} \) |
| 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 - 2.53T + 7T^{2} \) |
| 11 | \( 1 - 1.11T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 19 | \( 1 + 5.01T + 19T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 + 1.28T + 29T^{2} \) |
| 31 | \( 1 - 0.498T + 31T^{2} \) |
| 37 | \( 1 + 11.9T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 47 | \( 1 - 0.400T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 - 5.21T + 59T^{2} \) |
| 61 | \( 1 - 0.0130T + 61T^{2} \) |
| 67 | \( 1 + 8.24T + 67T^{2} \) |
| 71 | \( 1 + 8.15T + 71T^{2} \) |
| 73 | \( 1 - 5.99T + 73T^{2} \) |
| 79 | \( 1 + 1.43T + 79T^{2} \) |
| 83 | \( 1 + 1.09T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 6.18T + 97T^{2} \) |
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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
−10.25750350205496635939165222134, −9.026830485771565425339356669431, −8.282061455517674885545496171180, −7.32746757927295565065801806840, −6.26418066279004017806590276145, −5.33597653404152324270618841383, −4.55853515132601337156290026062, −3.88745321227365620147515252028, −1.36727566051240818908008943907, 0,
1.36727566051240818908008943907, 3.88745321227365620147515252028, 4.55853515132601337156290026062, 5.33597653404152324270618841383, 6.26418066279004017806590276145, 7.32746757927295565065801806840, 8.282061455517674885545496171180, 9.026830485771565425339356669431, 10.25750350205496635939165222134
