| L(s) = 1 | + 0.585·5-s − 7-s − 2.82·11-s − 5.65·13-s + 6.24·17-s + 19-s − 8.48·23-s − 4.65·25-s − 1.75·29-s + 3.17·31-s − 0.585·35-s − 6·37-s + 3.17·41-s − 3.17·43-s + 13.4·47-s + 49-s − 9.07·53-s − 1.65·55-s + 13.6·59-s + 2·61-s − 3.31·65-s − 14.8·67-s + 12.7·71-s − 3.17·73-s + 2.82·77-s + 13.6·79-s + 3.75·83-s + ⋯ |
| L(s) = 1 | + 0.261·5-s − 0.377·7-s − 0.852·11-s − 1.56·13-s + 1.51·17-s + 0.229·19-s − 1.76·23-s − 0.931·25-s − 0.326·29-s + 0.569·31-s − 0.0990·35-s − 0.986·37-s + 0.495·41-s − 0.483·43-s + 1.95·47-s + 0.142·49-s − 1.24·53-s − 0.223·55-s + 1.77·59-s + 0.256·61-s − 0.411·65-s − 1.81·67-s + 1.51·71-s − 0.371·73-s + 0.322·77-s + 1.53·79-s + 0.412·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.271506520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.271506520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 1.75T + 29T^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 3.17T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 3.75T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.76083065100596404967286202438, −7.15537255747868464387964204372, −6.22461875219590100730365318687, −5.56213161621931637802152680379, −5.13168951206594420682174749043, −4.16427336919708620066028792214, −3.39321886145384335730091767706, −2.53594682515733772089923285746, −1.90861539335780538358564877723, −0.51139164944939561271452412684,
0.51139164944939561271452412684, 1.90861539335780538358564877723, 2.53594682515733772089923285746, 3.39321886145384335730091767706, 4.16427336919708620066028792214, 5.13168951206594420682174749043, 5.56213161621931637802152680379, 6.22461875219590100730365318687, 7.15537255747868464387964204372, 7.76083065100596404967286202438
