| L(s) = 1 | + 2-s − 0.800·3-s + 4-s + 5-s − 0.800·6-s + 2.44·7-s + 8-s − 2.35·9-s + 10-s + 3.94·11-s − 0.800·12-s + 6.07·13-s + 2.44·14-s − 0.800·15-s + 16-s + 4.11·17-s − 2.35·18-s + 5.20·19-s + 20-s − 1.96·21-s + 3.94·22-s + 1.78·23-s − 0.800·24-s + 25-s + 6.07·26-s + 4.28·27-s + 2.44·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.462·3-s + 0.5·4-s + 0.447·5-s − 0.326·6-s + 0.925·7-s + 0.353·8-s − 0.786·9-s + 0.316·10-s + 1.18·11-s − 0.231·12-s + 1.68·13-s + 0.654·14-s − 0.206·15-s + 0.250·16-s + 0.999·17-s − 0.556·18-s + 1.19·19-s + 0.223·20-s − 0.427·21-s + 0.841·22-s + 0.371·23-s − 0.163·24-s + 0.200·25-s + 1.19·26-s + 0.825·27-s + 0.462·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.119400827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.119400827\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 - T \) |
| good | 3 | \( 1 + 0.800T + 3T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 - 3.94T + 11T^{2} \) |
| 13 | \( 1 - 6.07T + 13T^{2} \) |
| 17 | \( 1 - 4.11T + 17T^{2} \) |
| 19 | \( 1 - 5.20T + 19T^{2} \) |
| 23 | \( 1 - 1.78T + 23T^{2} \) |
| 29 | \( 1 + 5.89T + 29T^{2} \) |
| 31 | \( 1 - 3.62T + 31T^{2} \) |
| 37 | \( 1 - 0.432T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 3.34T + 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 - 0.615T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 - 4.14T + 67T^{2} \) |
| 71 | \( 1 - 4.18T + 71T^{2} \) |
| 73 | \( 1 + 2.73T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 - 0.343T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.062519106736068678681881078801, −7.22461459298871703796570841094, −6.37468640973769051921464699150, −5.84801637115751086539048634134, −5.34243480575978779710989542669, −4.56697808267272706937078799848, −3.58777289658793884161867745836, −3.07587309415355872729259909663, −1.63505158490205199663009825773, −1.14584457895502737341733298306,
1.14584457895502737341733298306, 1.63505158490205199663009825773, 3.07587309415355872729259909663, 3.58777289658793884161867745836, 4.56697808267272706937078799848, 5.34243480575978779710989542669, 5.84801637115751086539048634134, 6.37468640973769051921464699150, 7.22461459298871703796570841094, 8.062519106736068678681881078801
