L(s) = 1 | + 1.29·2-s − 2.64·3-s − 0.312·4-s − 3.43·6-s + 7-s − 3.00·8-s + 4.00·9-s + 6.40·11-s + 0.826·12-s + 2.78·13-s + 1.29·14-s − 3.27·16-s + 6.09·17-s + 5.19·18-s − 5.24·19-s − 2.64·21-s + 8.31·22-s + 23-s + 7.94·24-s + 3.62·26-s − 2.64·27-s − 0.312·28-s − 6.51·29-s − 6.15·31-s + 1.75·32-s − 16.9·33-s + 7.91·34-s + ⋯ |
L(s) = 1 | + 0.918·2-s − 1.52·3-s − 0.156·4-s − 1.40·6-s + 0.377·7-s − 1.06·8-s + 1.33·9-s + 1.92·11-s + 0.238·12-s + 0.773·13-s + 0.347·14-s − 0.819·16-s + 1.47·17-s + 1.22·18-s − 1.20·19-s − 0.577·21-s + 1.77·22-s + 0.208·23-s + 1.62·24-s + 0.710·26-s − 0.509·27-s − 0.0590·28-s − 1.21·29-s − 1.10·31-s + 0.309·32-s − 2.94·33-s + 1.35·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.696860071\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.696860071\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 1.29T + 2T^{2} \) |
| 3 | \( 1 + 2.64T + 3T^{2} \) |
| 11 | \( 1 - 6.40T + 11T^{2} \) |
| 13 | \( 1 - 2.78T + 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 19 | \( 1 + 5.24T + 19T^{2} \) |
| 29 | \( 1 + 6.51T + 29T^{2} \) |
| 31 | \( 1 + 6.15T + 31T^{2} \) |
| 37 | \( 1 + 6.68T + 37T^{2} \) |
| 41 | \( 1 - 0.0209T + 41T^{2} \) |
| 43 | \( 1 + 1.88T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 8.49T + 59T^{2} \) |
| 61 | \( 1 + 7.39T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 3.18T + 79T^{2} \) |
| 83 | \( 1 + 4.72T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−8.637061591107351380948935454861, −7.35069648116882346055719380015, −6.62950097884111154002000095668, −5.91376730787051732738652086207, −5.61239160258825435283783987497, −4.74981700491982516923933175168, −3.94042902934807812471645491183, −3.55231366613715260157390336405, −1.74832355979982010719576310558, −0.74174875977937740455301077034,
0.74174875977937740455301077034, 1.74832355979982010719576310558, 3.55231366613715260157390336405, 3.94042902934807812471645491183, 4.74981700491982516923933175168, 5.61239160258825435283783987497, 5.91376730787051732738652086207, 6.62950097884111154002000095668, 7.35069648116882346055719380015, 8.637061591107351380948935454861