L(s) = 1 | + 2-s − 3.23·3-s + 4-s + 5-s − 3.23·6-s − 7-s + 8-s + 7.47·9-s + 10-s − 3.23·12-s + 4.47·13-s − 14-s − 3.23·15-s + 16-s + 2·17-s + 7.47·18-s − 5.23·19-s + 20-s + 3.23·21-s − 7.23·23-s − 3.23·24-s + 25-s + 4.47·26-s − 14.4·27-s − 28-s − 1.23·29-s − 3.23·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.86·3-s + 0.5·4-s + 0.447·5-s − 1.32·6-s − 0.377·7-s + 0.353·8-s + 2.49·9-s + 0.316·10-s − 0.934·12-s + 1.24·13-s − 0.267·14-s − 0.835·15-s + 0.250·16-s + 0.485·17-s + 1.76·18-s − 1.20·19-s + 0.223·20-s + 0.706·21-s − 1.50·23-s − 0.660·24-s + 0.200·25-s + 0.877·26-s − 2.78·27-s − 0.188·28-s − 0.229·29-s − 0.590·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 + 7.23T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 0.472T + 47T^{2} \) |
| 53 | \( 1 + 0.763T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 1.70T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.92996313185405650257546706589, −6.44692416171195606953907312297, −6.03880807990291479605966724429, −5.54339967325523741573468428728, −4.73202350622291036944373109459, −4.13744539748480257134068059607, −3.35914745925531593028311679085, −2.00543099810267048338213191069, −1.22832229666786241891456387085, 0,
1.22832229666786241891456387085, 2.00543099810267048338213191069, 3.35914745925531593028311679085, 4.13744539748480257134068059607, 4.73202350622291036944373109459, 5.54339967325523741573468428728, 6.03880807990291479605966724429, 6.44692416171195606953907312297, 6.92996313185405650257546706589