| L(s) = 1 | + 2.17·2-s − 1.70·3-s + 2.70·4-s + 5-s − 3.70·6-s + 3.70·7-s + 1.53·8-s − 0.0783·9-s + 2.17·10-s − 0.630·11-s − 4.63·12-s − 4.34·13-s + 8.04·14-s − 1.70·15-s − 2.07·16-s − 1.55·17-s − 0.170·18-s − 5.70·19-s + 2.70·20-s − 6.34·21-s − 1.36·22-s + 6.63·23-s − 2.63·24-s + 25-s − 9.41·26-s + 5.26·27-s + 10.0·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 0.986·3-s + 1.35·4-s + 0.447·5-s − 1.51·6-s + 1.40·7-s + 0.544·8-s − 0.0261·9-s + 0.686·10-s − 0.190·11-s − 1.33·12-s − 1.20·13-s + 2.15·14-s − 0.441·15-s − 0.519·16-s − 0.376·17-s − 0.0400·18-s − 1.30·19-s + 0.605·20-s − 1.38·21-s − 0.291·22-s + 1.38·23-s − 0.537·24-s + 0.200·25-s − 1.84·26-s + 1.01·27-s + 1.89·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.890040786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.890040786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 3 | \( 1 + 1.70T + 3T^{2} \) |
| 7 | \( 1 - 3.70T + 7T^{2} \) |
| 11 | \( 1 + 0.630T + 11T^{2} \) |
| 13 | \( 1 + 4.34T + 13T^{2} \) |
| 17 | \( 1 + 1.55T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 - 6.63T + 23T^{2} \) |
| 31 | \( 1 + 2.29T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 2.29T + 47T^{2} \) |
| 53 | \( 1 - 0.921T + 53T^{2} \) |
| 59 | \( 1 + 3.60T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 - 1.41T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−12.92367278584047957304377174062, −12.27251964114220224436500218299, −11.23344300180097800016472423292, −10.75394280501581412525149868581, −8.924805684569023115861859221021, −7.29859012448595605091333569426, −6.07685665171122384653286800792, −5.13578476995828695758411370062, −4.52162632712868341357223355729, −2.40898589564752470898448557927,
2.40898589564752470898448557927, 4.52162632712868341357223355729, 5.13578476995828695758411370062, 6.07685665171122384653286800792, 7.29859012448595605091333569426, 8.924805684569023115861859221021, 10.75394280501581412525149868581, 11.23344300180097800016472423292, 12.27251964114220224436500218299, 12.92367278584047957304377174062
