| L(s) = 1 | + 2-s − 0.943·3-s + 4-s + 2.82·5-s − 0.943·6-s + 5.06·7-s + 8-s − 2.10·9-s + 2.82·10-s + 11-s − 0.943·12-s − 0.490·13-s + 5.06·14-s − 2.66·15-s + 16-s − 6.15·17-s − 2.10·18-s − 1.82·19-s + 2.82·20-s − 4.77·21-s + 22-s + 1.13·23-s − 0.943·24-s + 3.00·25-s − 0.490·26-s + 4.82·27-s + 5.06·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.544·3-s + 0.5·4-s + 1.26·5-s − 0.385·6-s + 1.91·7-s + 0.353·8-s − 0.703·9-s + 0.894·10-s + 0.301·11-s − 0.272·12-s − 0.135·13-s + 1.35·14-s − 0.689·15-s + 0.250·16-s − 1.49·17-s − 0.497·18-s − 0.418·19-s + 0.632·20-s − 1.04·21-s + 0.213·22-s + 0.236·23-s − 0.192·24-s + 0.601·25-s − 0.0961·26-s + 0.927·27-s + 0.957·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.948562972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.948562972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 0.943T + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 5.06T + 7T^{2} \) |
| 13 | \( 1 + 0.490T + 13T^{2} \) |
| 17 | \( 1 + 6.15T + 17T^{2} \) |
| 19 | \( 1 + 1.82T + 19T^{2} \) |
| 23 | \( 1 - 1.13T + 23T^{2} \) |
| 29 | \( 1 - 7.76T + 29T^{2} \) |
| 31 | \( 1 - 2.94T + 31T^{2} \) |
| 37 | \( 1 - 2.43T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 4.93T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 1.86T + 61T^{2} \) |
| 67 | \( 1 + 7.96T + 67T^{2} \) |
| 71 | \( 1 + 1.19T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 3.31T + 83T^{2} \) |
| 89 | \( 1 - 4.58T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
| 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
−8.590746819376613803141102595452, −7.53526110159316503162935301227, −6.55046734387912349904878411466, −6.12806744080531108012850590259, −5.26867512133330560716484185325, −4.84277514448712762976264180423, −4.17770999068346283943811911760, −2.61660731690675164102830939222, −2.11500426854290552461910678460, −1.11226675517114920428803915944,
1.11226675517114920428803915944, 2.11500426854290552461910678460, 2.61660731690675164102830939222, 4.17770999068346283943811911760, 4.84277514448712762976264180423, 5.26867512133330560716484185325, 6.12806744080531108012850590259, 6.55046734387912349904878411466, 7.53526110159316503162935301227, 8.590746819376613803141102595452
