| L(s) = 1 | − 1.39·2-s − 3-s − 0.0667·4-s + 2.45·5-s + 1.39·6-s + 7-s + 2.87·8-s + 9-s − 3.41·10-s − 2.83·11-s + 0.0667·12-s − 2.65·13-s − 1.39·14-s − 2.45·15-s − 3.86·16-s + 7.55·17-s − 1.39·18-s + 5.96·19-s − 0.164·20-s − 21-s + 3.94·22-s − 8.08·23-s − 2.87·24-s + 1.03·25-s + 3.68·26-s − 27-s − 0.0667·28-s + ⋯ |
| L(s) = 1 | − 0.983·2-s − 0.577·3-s − 0.0333·4-s + 1.09·5-s + 0.567·6-s + 0.377·7-s + 1.01·8-s + 0.333·9-s − 1.08·10-s − 0.856·11-s + 0.0192·12-s − 0.735·13-s − 0.371·14-s − 0.634·15-s − 0.965·16-s + 1.83·17-s − 0.327·18-s + 1.36·19-s − 0.0366·20-s − 0.218·21-s + 0.841·22-s − 1.68·23-s − 0.586·24-s + 0.207·25-s + 0.723·26-s − 0.192·27-s − 0.0126·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4767 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4767 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.034783732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.034783732\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 227 | \( 1 + T \) |
| good | 2 | \( 1 + 1.39T + 2T^{2} \) |
| 5 | \( 1 - 2.45T + 5T^{2} \) |
| 11 | \( 1 + 2.83T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 - 7.55T + 17T^{2} \) |
| 19 | \( 1 - 5.96T + 19T^{2} \) |
| 23 | \( 1 + 8.08T + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 + 8.81T + 31T^{2} \) |
| 37 | \( 1 + 0.252T + 37T^{2} \) |
| 41 | \( 1 - 0.867T + 41T^{2} \) |
| 43 | \( 1 - 6.62T + 43T^{2} \) |
| 47 | \( 1 + 8.86T + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 - 0.641T + 59T^{2} \) |
| 61 | \( 1 + 2.49T + 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 - 8.02T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 7.28T + 79T^{2} \) |
| 83 | \( 1 + 1.99T + 83T^{2} \) |
| 89 | \( 1 - 1.15T + 89T^{2} \) |
| 97 | \( 1 - 9.04T + 97T^{2} \) |
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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
−8.113941066006206983794950512028, −7.77391612091926358833750501389, −7.08412285779525655602252240392, −5.97569425735389814366565627522, −5.35288438788531307462299437538, −4.96104798408182030748502594326, −3.77338058925366975392220289983, −2.52114027791687969759129075542, −1.62029171441713465511965807491, −0.71021348385836382003379796534,
0.71021348385836382003379796534, 1.62029171441713465511965807491, 2.52114027791687969759129075542, 3.77338058925366975392220289983, 4.96104798408182030748502594326, 5.35288438788531307462299437538, 5.97569425735389814366565627522, 7.08412285779525655602252240392, 7.77391612091926358833750501389, 8.113941066006206983794950512028
