| L(s) = 1 | + 2-s + 3.25·3-s + 4-s − 0.0150·5-s + 3.25·6-s − 4.67·7-s + 8-s + 7.57·9-s − 0.0150·10-s + 1.23·11-s + 3.25·12-s + 4.71·13-s − 4.67·14-s − 0.0489·15-s + 16-s − 1.86·17-s + 7.57·18-s + 2.90·19-s − 0.0150·20-s − 15.2·21-s + 1.23·22-s + 5.87·23-s + 3.25·24-s − 4.99·25-s + 4.71·26-s + 14.8·27-s − 4.67·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.87·3-s + 0.5·4-s − 0.00672·5-s + 1.32·6-s − 1.76·7-s + 0.353·8-s + 2.52·9-s − 0.00475·10-s + 0.372·11-s + 0.938·12-s + 1.30·13-s − 1.25·14-s − 0.0126·15-s + 0.250·16-s − 0.452·17-s + 1.78·18-s + 0.667·19-s − 0.00336·20-s − 3.31·21-s + 0.263·22-s + 1.22·23-s + 0.663·24-s − 0.999·25-s + 0.925·26-s + 2.86·27-s − 0.883·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.742261204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.742261204\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 3.25T + 3T^{2} \) |
| 5 | \( 1 + 0.0150T + 5T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 - 4.71T + 13T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 19 | \( 1 - 2.90T + 19T^{2} \) |
| 23 | \( 1 - 5.87T + 23T^{2} \) |
| 29 | \( 1 - 1.80T + 29T^{2} \) |
| 31 | \( 1 + 5.87T + 31T^{2} \) |
| 37 | \( 1 + 6.28T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 2.88T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 3.59T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 1.97T + 61T^{2} \) |
| 67 | \( 1 + 1.89T + 67T^{2} \) |
| 71 | \( 1 + 6.21T + 71T^{2} \) |
| 73 | \( 1 + 0.840T + 73T^{2} \) |
| 79 | \( 1 + 0.938T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 - 9.17T + 97T^{2} \) |
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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
−8.684711605114361055952454420483, −7.56122659309574956911102059475, −7.04881551044646455342676604272, −6.38361457599619760969528472401, −5.52274046242503860287866080753, −4.07350935775877202200025792261, −3.77117932856076142928161302915, −3.07893122368641914214392343762, −2.44398133786871525782971317694, −1.23818536162665174688267755998,
1.23818536162665174688267755998, 2.44398133786871525782971317694, 3.07893122368641914214392343762, 3.77117932856076142928161302915, 4.07350935775877202200025792261, 5.52274046242503860287866080753, 6.38361457599619760969528472401, 7.04881551044646455342676604272, 7.56122659309574956911102059475, 8.684711605114361055952454420483
