| L(s) = 1 | + 2-s + 2.16·3-s + 4-s − 5-s + 2.16·6-s − 3.23·7-s + 8-s + 1.69·9-s − 10-s − 11-s + 2.16·12-s + 0.175·13-s − 3.23·14-s − 2.16·15-s + 16-s − 4.79·17-s + 1.69·18-s + 6.81·19-s − 20-s − 7.01·21-s − 22-s + 4.66·23-s + 2.16·24-s + 25-s + 0.175·26-s − 2.82·27-s − 3.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.25·3-s + 0.5·4-s − 0.447·5-s + 0.884·6-s − 1.22·7-s + 0.353·8-s + 0.565·9-s − 0.316·10-s − 0.301·11-s + 0.625·12-s + 0.0487·13-s − 0.865·14-s − 0.559·15-s + 0.250·16-s − 1.16·17-s + 0.399·18-s + 1.56·19-s − 0.223·20-s − 1.53·21-s − 0.213·22-s + 0.972·23-s + 0.442·24-s + 0.200·25-s + 0.0344·26-s − 0.543·27-s − 0.612·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 - 2.16T + 3T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 13 | \( 1 - 0.175T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 - 6.81T + 19T^{2} \) |
| 23 | \( 1 - 4.66T + 23T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 + 6.98T + 31T^{2} \) |
| 37 | \( 1 - 2.12T + 37T^{2} \) |
| 41 | \( 1 - 3.67T + 41T^{2} \) |
| 43 | \( 1 + 5.68T + 43T^{2} \) |
| 47 | \( 1 + 2.14T + 47T^{2} \) |
| 53 | \( 1 + 6.33T + 53T^{2} \) |
| 59 | \( 1 + 2.65T + 59T^{2} \) |
| 61 | \( 1 + 4.60T + 61T^{2} \) |
| 67 | \( 1 + 4.92T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 79 | \( 1 + 7.51T + 79T^{2} \) |
| 83 | \( 1 + 4.69T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 6.79T + 97T^{2} \) |
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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
−7.40794889351966841910883906016, −6.96950013208448918757902560003, −6.09887403138416217004578589444, −5.35023114994247425045861469421, −4.47152659680984035482366615257, −3.62817490221263668335242567721, −3.17237534843398380993095241638, −2.66427939758080314428705688182, −1.60621981019496050670129287446, 0,
1.60621981019496050670129287446, 2.66427939758080314428705688182, 3.17237534843398380993095241638, 3.62817490221263668335242567721, 4.47152659680984035482366615257, 5.35023114994247425045861469421, 6.09887403138416217004578589444, 6.96950013208448918757902560003, 7.40794889351966841910883906016
