| L(s) = 1 | − 0.803·2-s + 3-s − 1.35·4-s − 0.803·6-s + 0.508·7-s + 2.69·8-s + 9-s − 1.35·12-s + 5.13·13-s − 0.408·14-s + 0.544·16-s − 3.34·17-s − 0.803·18-s − 1.17·19-s + 0.508·21-s − 2.91·23-s + 2.69·24-s − 4.12·26-s + 27-s − 0.688·28-s − 0.392·29-s − 6.35·31-s − 5.82·32-s + 2.68·34-s − 1.35·36-s − 4.45·37-s + 0.942·38-s + ⋯ |
| L(s) = 1 | − 0.568·2-s + 0.577·3-s − 0.677·4-s − 0.327·6-s + 0.192·7-s + 0.952·8-s + 0.333·9-s − 0.391·12-s + 1.42·13-s − 0.109·14-s + 0.136·16-s − 0.811·17-s − 0.189·18-s − 0.269·19-s + 0.110·21-s − 0.607·23-s + 0.550·24-s − 0.809·26-s + 0.192·27-s − 0.130·28-s − 0.0729·29-s − 1.14·31-s − 1.03·32-s + 0.460·34-s − 0.225·36-s − 0.731·37-s + 0.152·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.803T + 2T^{2} \) |
| 7 | \( 1 - 0.508T + 7T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 17 | \( 1 + 3.34T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 2.91T + 23T^{2} \) |
| 29 | \( 1 + 0.392T + 29T^{2} \) |
| 31 | \( 1 + 6.35T + 31T^{2} \) |
| 37 | \( 1 + 4.45T + 37T^{2} \) |
| 41 | \( 1 + 2.79T + 41T^{2} \) |
| 43 | \( 1 - 3.05T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 9.80T + 53T^{2} \) |
| 59 | \( 1 + 3.02T + 59T^{2} \) |
| 61 | \( 1 - 1.05T + 61T^{2} \) |
| 67 | \( 1 + 5.31T + 67T^{2} \) |
| 71 | \( 1 - 4.09T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 9.84T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−7.49190826140585857966739547729, −6.98221713025558536575542653307, −5.97420906087544603459188093923, −5.38039624171330464476170309886, −4.23124039261664991606461343856, −4.06830617800330189903489614516, −3.08537110698972729967730467500, −1.95655609323412992105448959527, −1.26411503126427345189715375247, 0,
1.26411503126427345189715375247, 1.95655609323412992105448959527, 3.08537110698972729967730467500, 4.06830617800330189903489614516, 4.23124039261664991606461343856, 5.38039624171330464476170309886, 5.97420906087544603459188093923, 6.98221713025558536575542653307, 7.49190826140585857966739547729
