L(s) = 1 | + 0.0842·2-s − 1.99·4-s − 2.54·5-s − 1.62·7-s − 0.336·8-s − 0.214·10-s − 5.58·11-s + 1.97·13-s − 0.136·14-s + 3.95·16-s − 6.03·17-s − 3.46·19-s + 5.06·20-s − 0.470·22-s + 0.922·23-s + 1.46·25-s + 0.166·26-s + 3.23·28-s − 6.23·29-s + 0.733·31-s + 1.00·32-s − 0.508·34-s + 4.13·35-s − 5.69·37-s − 0.292·38-s + 0.855·40-s − 7.79·41-s + ⋯ |
L(s) = 1 | + 0.0595·2-s − 0.996·4-s − 1.13·5-s − 0.614·7-s − 0.118·8-s − 0.0677·10-s − 1.68·11-s + 0.549·13-s − 0.0365·14-s + 0.989·16-s − 1.46·17-s − 0.795·19-s + 1.13·20-s − 0.100·22-s + 0.192·23-s + 0.292·25-s + 0.0327·26-s + 0.612·28-s − 1.15·29-s + 0.131·31-s + 0.177·32-s − 0.0872·34-s + 0.698·35-s − 0.936·37-s − 0.0473·38-s + 0.135·40-s − 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3132131853\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3132131853\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
good | 2 | \( 1 - 0.0842T + 2T^{2} \) |
| 5 | \( 1 + 2.54T + 5T^{2} \) |
| 7 | \( 1 + 1.62T + 7T^{2} \) |
| 11 | \( 1 + 5.58T + 11T^{2} \) |
| 13 | \( 1 - 1.97T + 13T^{2} \) |
| 17 | \( 1 + 6.03T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 - 0.922T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 - 0.733T + 31T^{2} \) |
| 37 | \( 1 + 5.69T + 37T^{2} \) |
| 41 | \( 1 + 7.79T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 9.19T + 53T^{2} \) |
| 59 | \( 1 + 9.53T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 - 8.45T + 67T^{2} \) |
| 71 | \( 1 + 5.78T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 0.759T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 8.41T + 89T^{2} \) |
| 97 | \( 1 + 6.97T + 97T^{2} \) |
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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
−8.859994667658038441406269425890, −8.407994301241513499840226350945, −7.66998027767061490657045582218, −6.86026075952917373011424038249, −5.77365070291670195193163713265, −4.97351305382888260355363830369, −4.11513408432067462820331954306, −3.51998241456158742308772272664, −2.35926183220384618347775097837, −0.34386423185050138387792423129,
0.34386423185050138387792423129, 2.35926183220384618347775097837, 3.51998241456158742308772272664, 4.11513408432067462820331954306, 4.97351305382888260355363830369, 5.77365070291670195193163713265, 6.86026075952917373011424038249, 7.66998027767061490657045582218, 8.407994301241513499840226350945, 8.859994667658038441406269425890