| L(s) = 1 | + 1.13·2-s + 2.19·3-s − 0.711·4-s + 1.83·5-s + 2.49·6-s − 3.07·8-s + 1.81·9-s + 2.07·10-s + 4.13·11-s − 1.56·12-s + 4.73·13-s + 4.01·15-s − 2.07·16-s − 1.16·17-s + 2.06·18-s − 19-s − 1.30·20-s + 4.69·22-s − 0.834·23-s − 6.75·24-s − 1.64·25-s + 5.37·26-s − 2.59·27-s + 3.47·29-s + 4.56·30-s − 7.88·31-s + 3.80·32-s + ⋯ |
| L(s) = 1 | + 0.802·2-s + 1.26·3-s − 0.355·4-s + 0.818·5-s + 1.01·6-s − 1.08·8-s + 0.605·9-s + 0.657·10-s + 1.24·11-s − 0.450·12-s + 1.31·13-s + 1.03·15-s − 0.517·16-s − 0.283·17-s + 0.485·18-s − 0.229·19-s − 0.291·20-s + 1.00·22-s − 0.174·23-s − 1.37·24-s − 0.329·25-s + 1.05·26-s − 0.500·27-s + 0.645·29-s + 0.832·30-s − 1.41·31-s + 0.672·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.603643295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.603643295\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.13T + 2T^{2} \) |
| 3 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 - 1.83T + 5T^{2} \) |
| 11 | \( 1 - 4.13T + 11T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 17 | \( 1 + 1.16T + 17T^{2} \) |
| 23 | \( 1 + 0.834T + 23T^{2} \) |
| 29 | \( 1 - 3.47T + 29T^{2} \) |
| 31 | \( 1 + 7.88T + 31T^{2} \) |
| 37 | \( 1 - 9.51T + 37T^{2} \) |
| 41 | \( 1 - 2.27T + 41T^{2} \) |
| 43 | \( 1 - 8.77T + 43T^{2} \) |
| 47 | \( 1 + 4.49T + 47T^{2} \) |
| 53 | \( 1 + 7.62T + 53T^{2} \) |
| 59 | \( 1 + 3.26T + 59T^{2} \) |
| 61 | \( 1 + 7.85T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 8.72T + 71T^{2} \) |
| 73 | \( 1 + 7.69T + 73T^{2} \) |
| 79 | \( 1 + 6.90T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 3.40T + 89T^{2} \) |
| 97 | \( 1 + 3.04T + 97T^{2} \) |
| 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
−9.675030647299919606686203922943, −9.119308766009262918371270032452, −8.707506405586772519438598603672, −7.66239806589596022311389541001, −6.24921241294238134642608629947, −5.90442200757778733544529048911, −4.42742043288674521271924080101, −3.75874129122996296659290809427, −2.85210426903392303817413297017, −1.59209387417379272426075594376,
1.59209387417379272426075594376, 2.85210426903392303817413297017, 3.75874129122996296659290809427, 4.42742043288674521271924080101, 5.90442200757778733544529048911, 6.24921241294238134642608629947, 7.66239806589596022311389541001, 8.707506405586772519438598603672, 9.119308766009262918371270032452, 9.675030647299919606686203922943
