| L(s) = 1 | + 2·2-s + 5.92·3-s + 4·4-s + 5·5-s + 11.8·6-s + 24.6·7-s + 8·8-s + 8.13·9-s + 10·10-s − 24.6·11-s + 23.7·12-s + 15.2·13-s + 49.2·14-s + 29.6·15-s + 16·16-s − 78.7·17-s + 16.2·18-s − 16.3·19-s + 20·20-s + 145.·21-s − 49.3·22-s − 23·23-s + 47.4·24-s + 25·25-s + 30.5·26-s − 111.·27-s + 98.5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.14·3-s + 0.5·4-s + 0.447·5-s + 0.806·6-s + 1.32·7-s + 0.353·8-s + 0.301·9-s + 0.316·10-s − 0.675·11-s + 0.570·12-s + 0.325·13-s + 0.940·14-s + 0.510·15-s + 0.250·16-s − 1.12·17-s + 0.212·18-s − 0.197·19-s + 0.223·20-s + 1.51·21-s − 0.477·22-s − 0.208·23-s + 0.403·24-s + 0.200·25-s + 0.230·26-s − 0.797·27-s + 0.664·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.421407152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.421407152\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 5.92T + 27T^{2} \) |
| 7 | \( 1 - 24.6T + 343T^{2} \) |
| 11 | \( 1 + 24.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 16.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 36.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 186.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 394.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 252.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 8.12T + 1.48e5T^{2} \) |
| 59 | \( 1 - 173.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 420.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 142.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 658.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 144.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 521.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 987.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 176.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 769.T + 9.12e5T^{2} \) |
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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
−11.73275800189522106894910130970, −10.96182560532323280549541587766, −9.815756506694990984220477711761, −8.469561271869774254010460151693, −8.058416818496731504943099592534, −6.66078305501590658723571797001, −5.29323358565995430097231771869, −4.28133556753642044317321175150, −2.80147811008455686815916921002, −1.82426329854911337124867024614,
1.82426329854911337124867024614, 2.80147811008455686815916921002, 4.28133556753642044317321175150, 5.29323358565995430097231771869, 6.66078305501590658723571797001, 8.058416818496731504943099592534, 8.469561271869774254010460151693, 9.815756506694990984220477711761, 10.96182560532323280549541587766, 11.73275800189522106894910130970
