| L(s) = 1 | + 2.57·2-s − 1.38·3-s + 4.65·4-s + 5-s − 3.57·6-s + 7-s + 6.84·8-s − 1.07·9-s + 2.57·10-s + 4.26·11-s − 6.45·12-s − 1.57·13-s + 2.57·14-s − 1.38·15-s + 8.35·16-s + 17-s − 2.77·18-s − 4.26·19-s + 4.65·20-s − 1.38·21-s + 11.0·22-s − 0.266·23-s − 9.49·24-s + 25-s − 4.07·26-s + 5.65·27-s + 4.65·28-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 0.801·3-s + 2.32·4-s + 0.447·5-s − 1.46·6-s + 0.377·7-s + 2.42·8-s − 0.358·9-s + 0.815·10-s + 1.28·11-s − 1.86·12-s − 0.438·13-s + 0.689·14-s − 0.358·15-s + 2.08·16-s + 0.242·17-s − 0.653·18-s − 0.978·19-s + 1.04·20-s − 0.302·21-s + 2.34·22-s − 0.0555·23-s − 1.93·24-s + 0.200·25-s − 0.799·26-s + 1.08·27-s + 0.879·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.637317134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.637317134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 2.57T + 2T^{2} \) |
| 3 | \( 1 + 1.38T + 3T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 19 | \( 1 + 4.26T + 19T^{2} \) |
| 23 | \( 1 + 0.266T + 23T^{2} \) |
| 29 | \( 1 + 2.95T + 29T^{2} \) |
| 31 | \( 1 + 0.583T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 + 3.57T + 43T^{2} \) |
| 47 | \( 1 + 5.88T + 47T^{2} \) |
| 53 | \( 1 + 7.73T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 3.50T + 61T^{2} \) |
| 67 | \( 1 + 3.26T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 0.191T + 73T^{2} \) |
| 79 | \( 1 - 0.163T + 79T^{2} \) |
| 83 | \( 1 + 4.00T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 8.63T + 97T^{2} \) |
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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
−11.22114656988778006220677393455, −10.23967355167450759564995024726, −8.969642580907784668093309942472, −7.56984674163711425299829863294, −6.34967438114348761758365760202, −6.14794992768772779723630677505, −5.03520951674539196675443870236, −4.36010985067546135055976024154, −3.13324929943694919937347053137, −1.78399131498275490911683526285,
1.78399131498275490911683526285, 3.13324929943694919937347053137, 4.36010985067546135055976024154, 5.03520951674539196675443870236, 6.14794992768772779723630677505, 6.34967438114348761758365760202, 7.56984674163711425299829863294, 8.969642580907784668093309942472, 10.23967355167450759564995024726, 11.22114656988778006220677393455
