L(s) = 1 | − 2-s + 1.20·3-s + 4-s + 2.17·5-s − 1.20·6-s + 7-s − 8-s − 1.54·9-s − 2.17·10-s − 1.32·11-s + 1.20·12-s − 6.95·13-s − 14-s + 2.61·15-s + 16-s + 5.76·17-s + 1.54·18-s − 3.84·19-s + 2.17·20-s + 1.20·21-s + 1.32·22-s − 2.41·23-s − 1.20·24-s − 0.286·25-s + 6.95·26-s − 5.48·27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.696·3-s + 0.5·4-s + 0.970·5-s − 0.492·6-s + 0.377·7-s − 0.353·8-s − 0.515·9-s − 0.686·10-s − 0.399·11-s + 0.348·12-s − 1.92·13-s − 0.267·14-s + 0.676·15-s + 0.250·16-s + 1.39·17-s + 0.364·18-s − 0.882·19-s + 0.485·20-s + 0.263·21-s + 0.282·22-s − 0.502·23-s − 0.246·24-s − 0.0572·25-s + 1.36·26-s − 1.05·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 - 1.20T + 3T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 + 6.95T + 13T^{2} \) |
| 17 | \( 1 - 5.76T + 17T^{2} \) |
| 19 | \( 1 + 3.84T + 19T^{2} \) |
| 23 | \( 1 + 2.41T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 9.33T + 31T^{2} \) |
| 37 | \( 1 - 5.03T + 37T^{2} \) |
| 41 | \( 1 - 2.43T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 5.31T + 53T^{2} \) |
| 59 | \( 1 - 6.17T + 59T^{2} \) |
| 61 | \( 1 + 7.19T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 + 8.33T + 71T^{2} \) |
| 73 | \( 1 - 5.89T + 73T^{2} \) |
| 79 | \( 1 + 6.68T + 79T^{2} \) |
| 83 | \( 1 - 5.71T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 9.37T + 97T^{2} \) |
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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
−7.85889928064333122079333160842, −7.30338642746301742575682556402, −6.36111303698857877567930435627, −5.57315376130094779176786464476, −5.06185865616799025400991328317, −3.90976240878098516121683535139, −2.61408191077566149404711631456, −2.52376520470006870206893663622, −1.48126606591353479391990498375, 0,
1.48126606591353479391990498375, 2.52376520470006870206893663622, 2.61408191077566149404711631456, 3.90976240878098516121683535139, 5.06185865616799025400991328317, 5.57315376130094779176786464476, 6.36111303698857877567930435627, 7.30338642746301742575682556402, 7.85889928064333122079333160842