| L(s) = 1 | − 2-s + 2.94·3-s + 4-s − 3.36·5-s − 2.94·6-s − 1.48·7-s − 8-s + 5.67·9-s + 3.36·10-s + 5.38·11-s + 2.94·12-s − 4.75·13-s + 1.48·14-s − 9.92·15-s + 16-s − 4.19·17-s − 5.67·18-s − 1.48·19-s − 3.36·20-s − 4.38·21-s − 5.38·22-s − 3.50·23-s − 2.94·24-s + 6.34·25-s + 4.75·26-s + 7.88·27-s − 1.48·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.70·3-s + 0.5·4-s − 1.50·5-s − 1.20·6-s − 0.562·7-s − 0.353·8-s + 1.89·9-s + 1.06·10-s + 1.62·11-s + 0.850·12-s − 1.31·13-s + 0.397·14-s − 2.56·15-s + 0.250·16-s − 1.01·17-s − 1.33·18-s − 0.341·19-s − 0.753·20-s − 0.956·21-s − 1.14·22-s − 0.730·23-s − 0.601·24-s + 1.26·25-s + 0.933·26-s + 1.51·27-s − 0.281·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 2.94T + 3T^{2} \) |
| 5 | \( 1 + 3.36T + 5T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 + 4.75T + 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 + 3.50T + 23T^{2} \) |
| 29 | \( 1 - 6.24T + 29T^{2} \) |
| 31 | \( 1 - 7.89T + 31T^{2} \) |
| 37 | \( 1 + 7.02T + 37T^{2} \) |
| 41 | \( 1 + 0.167T + 41T^{2} \) |
| 43 | \( 1 - 9.71T + 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 53 | \( 1 + 7.64T + 53T^{2} \) |
| 59 | \( 1 + 4.74T + 59T^{2} \) |
| 61 | \( 1 - 1.66T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 2.45T + 79T^{2} \) |
| 83 | \( 1 - 9.62T + 83T^{2} \) |
| 89 | \( 1 + 4.74T + 89T^{2} \) |
| 97 | \( 1 + 6.68T + 97T^{2} \) |
| 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
−8.159914495891847773615899326713, −7.60028999882461117066077587658, −6.90418646835876786804117218494, −6.41951937373900101139364154351, −4.44909155939389486819211496231, −4.19659690631915436753250007952, −3.20827086332958688698254141283, −2.63667824856079628051122155160, −1.49820058152203364067649600978, 0,
1.49820058152203364067649600978, 2.63667824856079628051122155160, 3.20827086332958688698254141283, 4.19659690631915436753250007952, 4.44909155939389486819211496231, 6.41951937373900101139364154351, 6.90418646835876786804117218494, 7.60028999882461117066077587658, 8.159914495891847773615899326713
