L(s) = 1 | − 0.595·2-s − 0.809·3-s − 1.64·4-s − 5-s + 0.482·6-s − 4.26·7-s + 2.17·8-s − 2.34·9-s + 0.595·10-s + 11-s + 1.33·12-s − 3.12·13-s + 2.54·14-s + 0.809·15-s + 1.99·16-s + 2.46·17-s + 1.39·18-s + 2.20·19-s + 1.64·20-s + 3.45·21-s − 0.595·22-s − 0.564·23-s − 1.75·24-s + 25-s + 1.86·26-s + 4.32·27-s + 7.02·28-s + ⋯ |
L(s) = 1 | − 0.421·2-s − 0.467·3-s − 0.822·4-s − 0.447·5-s + 0.196·6-s − 1.61·7-s + 0.767·8-s − 0.781·9-s + 0.188·10-s + 0.301·11-s + 0.384·12-s − 0.866·13-s + 0.679·14-s + 0.208·15-s + 0.498·16-s + 0.596·17-s + 0.329·18-s + 0.505·19-s + 0.367·20-s + 0.753·21-s − 0.127·22-s − 0.117·23-s − 0.358·24-s + 0.200·25-s + 0.364·26-s + 0.832·27-s + 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 + 0.595T + 2T^{2} \) |
| 3 | \( 1 + 0.809T + 3T^{2} \) |
| 7 | \( 1 + 4.26T + 7T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 - 2.20T + 19T^{2} \) |
| 23 | \( 1 + 0.564T + 23T^{2} \) |
| 29 | \( 1 + 5.17T + 29T^{2} \) |
| 31 | \( 1 - 8.87T + 31T^{2} \) |
| 37 | \( 1 + 2.41T + 37T^{2} \) |
| 41 | \( 1 - 6.48T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 9.33T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 5.90T + 59T^{2} \) |
| 61 | \( 1 - 2.22T + 61T^{2} \) |
| 67 | \( 1 - 0.968T + 67T^{2} \) |
| 71 | \( 1 - 2.24T + 71T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 - 6.31T + 83T^{2} \) |
| 89 | \( 1 + 1.82T + 89T^{2} \) |
| 97 | \( 1 + 1.63T + 97T^{2} \) |
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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
−8.188213262427111189508025069294, −7.34628802389973079148365202604, −6.69421119214160165243918181805, −5.77423786535092532136831923575, −5.22486755003551283443537793877, −4.19486199183030617410828428369, −3.45244760439671236747592324592, −2.63924439315818796752735833564, −0.858006492041433291130812407176, 0,
0.858006492041433291130812407176, 2.63924439315818796752735833564, 3.45244760439671236747592324592, 4.19486199183030617410828428369, 5.22486755003551283443537793877, 5.77423786535092532136831923575, 6.69421119214160165243918181805, 7.34628802389973079148365202604, 8.188213262427111189508025069294