| L(s) = 1 | + 0.218·2-s − 1.19·3-s − 1.95·4-s + 1.50·5-s − 0.260·6-s + 3.58·7-s − 0.863·8-s − 1.57·9-s + 0.328·10-s − 3.95·11-s + 2.32·12-s + 1.73·13-s + 0.782·14-s − 1.79·15-s + 3.71·16-s + 1.71·17-s − 0.344·18-s − 5.14·19-s − 2.93·20-s − 4.26·21-s − 0.863·22-s + 4.66·23-s + 1.02·24-s − 2.73·25-s + 0.379·26-s + 5.45·27-s − 6.98·28-s + ⋯ |
| L(s) = 1 | + 0.154·2-s − 0.688·3-s − 0.976·4-s + 0.672·5-s − 0.106·6-s + 1.35·7-s − 0.305·8-s − 0.526·9-s + 0.103·10-s − 1.19·11-s + 0.671·12-s + 0.481·13-s + 0.209·14-s − 0.463·15-s + 0.928·16-s + 0.416·17-s − 0.0812·18-s − 1.17·19-s − 0.656·20-s − 0.931·21-s − 0.184·22-s + 0.973·23-s + 0.210·24-s − 0.547·25-s + 0.0743·26-s + 1.05·27-s − 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2671 | \( 1 + T \) |
| good | 2 | \( 1 - 0.218T + 2T^{2} \) |
| 3 | \( 1 + 1.19T + 3T^{2} \) |
| 5 | \( 1 - 1.50T + 5T^{2} \) |
| 7 | \( 1 - 3.58T + 7T^{2} \) |
| 11 | \( 1 + 3.95T + 11T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 - 1.71T + 17T^{2} \) |
| 19 | \( 1 + 5.14T + 19T^{2} \) |
| 23 | \( 1 - 4.66T + 23T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 + 7.24T + 31T^{2} \) |
| 37 | \( 1 + 7.27T + 37T^{2} \) |
| 41 | \( 1 + 3.30T + 41T^{2} \) |
| 43 | \( 1 - 4.39T + 43T^{2} \) |
| 47 | \( 1 + 8.49T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 1.15T + 59T^{2} \) |
| 61 | \( 1 + 2.20T + 61T^{2} \) |
| 67 | \( 1 - 1.95T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 8.55T + 79T^{2} \) |
| 83 | \( 1 + 1.32T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−8.419248844310976279974877934499, −7.997480750785336886661816611504, −6.83202455847634059217034793881, −5.74001874505911888612048293246, −5.33982940129554726843771041724, −4.85129513215881560764956362058, −3.81639274938678544821531613515, −2.57763145517012814518275030751, −1.41068274408165552826855146166, 0,
1.41068274408165552826855146166, 2.57763145517012814518275030751, 3.81639274938678544821531613515, 4.85129513215881560764956362058, 5.33982940129554726843771041724, 5.74001874505911888612048293246, 6.83202455847634059217034793881, 7.997480750785336886661816611504, 8.419248844310976279974877934499
