| L(s) = 1 | − 2-s + 4-s − 2.70·5-s + 1.70·7-s − 8-s + 2.70·10-s + 1.70·11-s + 4.70·13-s − 1.70·14-s + 16-s + 2·17-s + 4·19-s − 2.70·20-s − 1.70·22-s + 2.29·25-s − 4.70·26-s + 1.70·28-s − 6.40·29-s − 5.70·31-s − 32-s − 2·34-s − 4.59·35-s − 9.40·37-s − 4·38-s + 2.70·40-s + 10.1·41-s − 11.4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.20·5-s + 0.643·7-s − 0.353·8-s + 0.854·10-s + 0.513·11-s + 1.30·13-s − 0.454·14-s + 0.250·16-s + 0.485·17-s + 0.917·19-s − 0.604·20-s − 0.362·22-s + 0.459·25-s − 0.922·26-s + 0.321·28-s − 1.18·29-s − 1.02·31-s − 0.176·32-s − 0.342·34-s − 0.777·35-s − 1.54·37-s − 0.648·38-s + 0.427·40-s + 1.57·41-s − 1.73·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 - 1.70T + 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 - 4.70T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 + 6.40T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 6.40T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 7.29T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 6.40T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 6.29T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−7.51563840228941624728871270128, −6.96802834335738675221461495458, −6.04536573022023476450542804158, −5.38537574104037840520496152654, −4.46336694387098808467792677829, −3.56910698364806009697694896681, −3.32588202870484440058590542970, −1.82391311859093913980914310422, −1.19290978172597770755914330946, 0,
1.19290978172597770755914330946, 1.82391311859093913980914310422, 3.32588202870484440058590542970, 3.56910698364806009697694896681, 4.46336694387098808467792677829, 5.38537574104037840520496152654, 6.04536573022023476450542804158, 6.96802834335738675221461495458, 7.51563840228941624728871270128
