| L(s) = 1 | − 2-s − 3-s + 4-s + 3.29·5-s + 6-s + 3.95·7-s − 8-s + 9-s − 3.29·10-s − 4.31·11-s − 12-s + 1.23·13-s − 3.95·14-s − 3.29·15-s + 16-s + 5.14·17-s − 18-s − 1.23·19-s + 3.29·20-s − 3.95·21-s + 4.31·22-s − 6·23-s + 24-s + 5.85·25-s − 1.23·26-s − 27-s + 3.95·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.47·5-s + 0.408·6-s + 1.49·7-s − 0.353·8-s + 0.333·9-s − 1.04·10-s − 1.30·11-s − 0.288·12-s + 0.342·13-s − 1.05·14-s − 0.850·15-s + 0.250·16-s + 1.24·17-s − 0.235·18-s − 0.283·19-s + 0.736·20-s − 0.863·21-s + 0.920·22-s − 1.25·23-s + 0.204·24-s + 1.17·25-s − 0.242·26-s − 0.192·27-s + 0.747·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.839311051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.839311051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 - 3.29T + 5T^{2} \) |
| 7 | \( 1 - 3.95T + 7T^{2} \) |
| 11 | \( 1 + 4.31T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 5.14T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 37 | \( 1 - 1.02T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 + 1.37T + 47T^{2} \) |
| 53 | \( 1 - 6.85T + 53T^{2} \) |
| 59 | \( 1 - 7.99T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 7.98T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 5.63T + 83T^{2} \) |
| 89 | \( 1 - 0.0629T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−8.223420741425600145854385075875, −7.54567575605281850571062616828, −6.69004161517478661395327708433, −5.82551289492391540864901711402, −5.38688061744479567663898307296, −4.88792719630879162917259115351, −3.57886679668895394342288701885, −2.22445799196257869673340806620, −1.88632680429903138382513631513, −0.847255571622085700002176165937,
0.847255571622085700002176165937, 1.88632680429903138382513631513, 2.22445799196257869673340806620, 3.57886679668895394342288701885, 4.88792719630879162917259115351, 5.38688061744479567663898307296, 5.82551289492391540864901711402, 6.69004161517478661395327708433, 7.54567575605281850571062616828, 8.223420741425600145854385075875
