| L(s) = 1 | − 2.32·2-s + 2.00·3-s + 3.40·4-s + 3.04·5-s − 4.65·6-s − 5.08·7-s − 3.26·8-s + 1.00·9-s − 7.08·10-s + 11-s + 6.81·12-s + 11.8·14-s + 6.09·15-s + 0.781·16-s + 4.98·17-s − 2.33·18-s − 4.25·19-s + 10.3·20-s − 10.1·21-s − 2.32·22-s + 3.33·23-s − 6.53·24-s + 4.28·25-s − 3.99·27-s − 17.3·28-s + 0.0575·29-s − 14.1·30-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 1.15·3-s + 1.70·4-s + 1.36·5-s − 1.89·6-s − 1.92·7-s − 1.15·8-s + 0.334·9-s − 2.24·10-s + 0.301·11-s + 1.96·12-s + 3.16·14-s + 1.57·15-s + 0.195·16-s + 1.20·17-s − 0.550·18-s − 0.976·19-s + 2.32·20-s − 2.22·21-s − 0.495·22-s + 0.696·23-s − 1.33·24-s + 0.857·25-s − 0.768·27-s − 3.27·28-s + 0.0106·29-s − 2.58·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223728405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223728405\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 - 2.00T + 3T^{2} \) |
| 5 | \( 1 - 3.04T + 5T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 17 | \( 1 - 4.98T + 17T^{2} \) |
| 19 | \( 1 + 4.25T + 19T^{2} \) |
| 23 | \( 1 - 3.33T + 23T^{2} \) |
| 29 | \( 1 - 0.0575T + 29T^{2} \) |
| 31 | \( 1 - 4.65T + 31T^{2} \) |
| 37 | \( 1 + 1.03T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 0.344T + 43T^{2} \) |
| 47 | \( 1 - 5.15T + 47T^{2} \) |
| 53 | \( 1 - 5.27T + 53T^{2} \) |
| 59 | \( 1 + 4.35T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 0.194T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 9.71T + 73T^{2} \) |
| 79 | \( 1 + 3.22T + 79T^{2} \) |
| 83 | \( 1 - 7.54T + 83T^{2} \) |
| 89 | \( 1 + 2.89T + 89T^{2} \) |
| 97 | \( 1 + 3.31T + 97T^{2} \) |
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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
−9.300452314347684814566844973158, −8.793894217766409976560063684617, −7.950508349927862239017701455667, −7.01636065491250999243416674988, −6.38539614443282441812733384406, −5.67326640273837019848820283424, −3.80127958556579066820134048143, −2.75831156016787631291471612967, −2.28088928231353786887212719366, −0.903344307775768006244395466344,
0.903344307775768006244395466344, 2.28088928231353786887212719366, 2.75831156016787631291471612967, 3.80127958556579066820134048143, 5.67326640273837019848820283424, 6.38539614443282441812733384406, 7.01636065491250999243416674988, 7.950508349927862239017701455667, 8.793894217766409976560063684617, 9.300452314347684814566844973158
