| L(s) = 1 | + 2.52·2-s − 3-s + 4.39·4-s + 0.133·5-s − 2.52·6-s − 7-s + 6.05·8-s + 9-s + 0.337·10-s + 11-s − 4.39·12-s + 0.133·13-s − 2.52·14-s − 0.133·15-s + 6.52·16-s − 5.05·17-s + 2.52·18-s − 0.924·19-s + 0.586·20-s + 21-s + 2.52·22-s − 7.05·23-s − 6.05·24-s − 4.98·25-s + 0.337·26-s − 27-s − 4.39·28-s + ⋯ |
| L(s) = 1 | + 1.78·2-s − 0.577·3-s + 2.19·4-s + 0.0596·5-s − 1.03·6-s − 0.377·7-s + 2.14·8-s + 0.333·9-s + 0.106·10-s + 0.301·11-s − 1.26·12-s + 0.0370·13-s − 0.675·14-s − 0.0344·15-s + 1.63·16-s − 1.22·17-s + 0.596·18-s − 0.212·19-s + 0.131·20-s + 0.218·21-s + 0.539·22-s − 1.47·23-s − 1.23·24-s − 0.996·25-s + 0.0662·26-s − 0.192·27-s − 0.830·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.635423434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.635423434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 5 | \( 1 - 0.133T + 5T^{2} \) |
| 13 | \( 1 - 0.133T + 13T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + 0.924T + 19T^{2} \) |
| 23 | \( 1 + 7.05T + 23T^{2} \) |
| 29 | \( 1 - 3.86T + 29T^{2} \) |
| 31 | \( 1 - 2.79T + 31T^{2} \) |
| 37 | \( 1 - 9.98T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 3.05T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 + 4.79T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 8.92T + 67T^{2} \) |
| 71 | \( 1 + 6.11T + 71T^{2} \) |
| 73 | \( 1 - 7.86T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 1.20T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−12.31939094116399208259155120933, −11.58005232048979331673763077630, −10.77921814662342343046734326351, −9.558770604813578568531728217260, −7.81193716649814418758385755456, −6.43847728305086804383472755460, −6.08176024111541615510085979932, −4.71114374854496168174718887749, −3.89767349904112176684886328824, −2.32788722497410658387544301182,
2.32788722497410658387544301182, 3.89767349904112176684886328824, 4.71114374854496168174718887749, 6.08176024111541615510085979932, 6.43847728305086804383472755460, 7.81193716649814418758385755456, 9.558770604813578568531728217260, 10.77921814662342343046734326351, 11.58005232048979331673763077630, 12.31939094116399208259155120933
