| L(s) = 1 | + 2-s + 1.34·3-s + 4-s − 2.31·5-s + 1.34·6-s + 7-s + 8-s − 1.19·9-s − 2.31·10-s + 1.82·11-s + 1.34·12-s + 1.02·13-s + 14-s − 3.11·15-s + 16-s − 6.30·17-s − 1.19·18-s + 3.88·19-s − 2.31·20-s + 1.34·21-s + 1.82·22-s − 7.18·23-s + 1.34·24-s + 0.360·25-s + 1.02·26-s − 5.63·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.776·3-s + 0.5·4-s − 1.03·5-s + 0.549·6-s + 0.377·7-s + 0.353·8-s − 0.396·9-s − 0.732·10-s + 0.549·11-s + 0.388·12-s + 0.284·13-s + 0.267·14-s − 0.804·15-s + 0.250·16-s − 1.52·17-s − 0.280·18-s + 0.890·19-s − 0.517·20-s + 0.293·21-s + 0.388·22-s − 1.49·23-s + 0.274·24-s + 0.0720·25-s + 0.201·26-s − 1.08·27-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 | \( 1 - 1.34T + 3T^{2} \) |
| 5 | \( 1 + 2.31T + 5T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 - 1.02T + 13T^{2} \) |
| 17 | \( 1 + 6.30T + 17T^{2} \) |
| 19 | \( 1 - 3.88T + 19T^{2} \) |
| 23 | \( 1 + 7.18T + 23T^{2} \) |
| 29 | \( 1 + 4.35T + 29T^{2} \) |
| 31 | \( 1 - 0.296T + 31T^{2} \) |
| 37 | \( 1 + 4.26T + 37T^{2} \) |
| 41 | \( 1 + 5.48T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 0.609T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 + 9.71T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 + 2.69T + 71T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 + 2.85T + 79T^{2} \) |
| 83 | \( 1 + 0.509T + 83T^{2} \) |
| 89 | \( 1 + 5.16T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.75389712655943435484233427377, −7.13159999986655304948688592033, −6.26090543540989470756043055850, −5.54980365203488608371128586769, −4.55554001862028228384323255317, −3.94491632078752319785872552195, −3.44903703879966819566450560769, −2.48694471054501540159293494072, −1.65864735629276543386993588679, 0,
1.65864735629276543386993588679, 2.48694471054501540159293494072, 3.44903703879966819566450560769, 3.94491632078752319785872552195, 4.55554001862028228384323255317, 5.54980365203488608371128586769, 6.26090543540989470756043055850, 7.13159999986655304948688592033, 7.75389712655943435484233427377
