| L(s) = 1 | − 0.0885·2-s − 1.99·4-s + 5-s − 7-s + 0.353·8-s − 0.0885·10-s − 0.355·11-s + 2.85·13-s + 0.0885·14-s + 3.95·16-s − 2.88·17-s + 1.92·19-s − 1.99·20-s + 0.0315·22-s − 5.38·23-s + 25-s − 0.252·26-s + 1.99·28-s − 29-s − 2.71·31-s − 1.05·32-s + 0.255·34-s − 35-s − 1.04·37-s − 0.170·38-s + 0.353·40-s + 1.42·41-s + ⋯ |
| L(s) = 1 | − 0.0626·2-s − 0.996·4-s + 0.447·5-s − 0.377·7-s + 0.124·8-s − 0.0280·10-s − 0.107·11-s + 0.791·13-s + 0.0236·14-s + 0.988·16-s − 0.699·17-s + 0.442·19-s − 0.445·20-s + 0.00671·22-s − 1.12·23-s + 0.200·25-s − 0.0495·26-s + 0.376·28-s − 0.185·29-s − 0.488·31-s − 0.186·32-s + 0.0438·34-s − 0.169·35-s − 0.171·37-s − 0.0277·38-s + 0.0558·40-s + 0.223·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0885T + 2T^{2} \) |
| 11 | \( 1 + 0.355T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 + 2.88T + 17T^{2} \) |
| 19 | \( 1 - 1.92T + 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 31 | \( 1 + 2.71T + 31T^{2} \) |
| 37 | \( 1 + 1.04T + 37T^{2} \) |
| 41 | \( 1 - 1.42T + 41T^{2} \) |
| 43 | \( 1 - 4.44T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 6.06T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 - 6.07T + 61T^{2} \) |
| 67 | \( 1 + 2.52T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 7.19T + 73T^{2} \) |
| 79 | \( 1 + 5.80T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 7.86T + 89T^{2} \) |
| 97 | \( 1 + 2.82T + 97T^{2} \) |
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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
−7.50166853857269436858463394276, −6.60041723499804687865755182740, −5.89327740312760815451146650964, −5.44031043616382287677809418045, −4.49357296593642148843531698959, −3.93352639880785144885048067696, −3.16728044362583480521539509188, −2.13867777846407812572696792816, −1.12529962333764293457811499577, 0,
1.12529962333764293457811499577, 2.13867777846407812572696792816, 3.16728044362583480521539509188, 3.93352639880785144885048067696, 4.49357296593642148843531698959, 5.44031043616382287677809418045, 5.89327740312760815451146650964, 6.60041723499804687865755182740, 7.50166853857269436858463394276
