| L(s) = 1 | − 2.14·2-s + 2.59·4-s − 5-s − 1.53·7-s − 1.27·8-s + 2.14·10-s − 11-s − 1.14·13-s + 3.29·14-s − 2.45·16-s − 3.14·17-s + 19-s − 2.59·20-s + 2.14·22-s + 5.10·23-s + 25-s + 2.45·26-s − 3.98·28-s − 2.99·29-s + 0.460·31-s + 7.81·32-s + 6.74·34-s + 1.53·35-s − 7.62·37-s − 2.14·38-s + 1.27·40-s + 1.79·41-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 1.29·4-s − 0.447·5-s − 0.580·7-s − 0.450·8-s + 0.677·10-s − 0.301·11-s − 0.317·13-s + 0.879·14-s − 0.614·16-s − 0.762·17-s + 0.229·19-s − 0.580·20-s + 0.456·22-s + 1.06·23-s + 0.200·25-s + 0.481·26-s − 0.752·28-s − 0.556·29-s + 0.0827·31-s + 1.38·32-s + 1.15·34-s + 0.259·35-s − 1.25·37-s − 0.347·38-s + 0.201·40-s + 0.280·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 13 | \( 1 + 1.14T + 13T^{2} \) |
| 17 | \( 1 + 3.14T + 17T^{2} \) |
| 23 | \( 1 - 5.10T + 23T^{2} \) |
| 29 | \( 1 + 2.99T + 29T^{2} \) |
| 31 | \( 1 - 0.460T + 31T^{2} \) |
| 37 | \( 1 + 7.62T + 37T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 0.0901T + 47T^{2} \) |
| 53 | \( 1 + 2.98T + 53T^{2} \) |
| 59 | \( 1 - 7.98T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 9.20T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 4.27T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−7.43399010756401432432975473514, −6.97768306421645791196465566894, −6.37920855863236867739757126426, −5.34894813617305951356075150396, −4.59778892218138200158884974455, −3.66759039019303880854685009147, −2.77982209120409767736875232490, −1.99753903194109081320447012897, −0.884428978131899938845889791585, 0,
0.884428978131899938845889791585, 1.99753903194109081320447012897, 2.77982209120409767736875232490, 3.66759039019303880854685009147, 4.59778892218138200158884974455, 5.34894813617305951356075150396, 6.37920855863236867739757126426, 6.97768306421645791196465566894, 7.43399010756401432432975473514
