| L(s) = 1 | + 2-s − 0.314·3-s + 4-s + 3.46·5-s − 0.314·6-s − 0.478·7-s + 8-s − 2.90·9-s + 3.46·10-s + 2.30·11-s − 0.314·12-s + 1.35·13-s − 0.478·14-s − 1.08·15-s + 16-s − 0.642·17-s − 2.90·18-s + 1.56·19-s + 3.46·20-s + 0.150·21-s + 2.30·22-s + 2.29·23-s − 0.314·24-s + 7.00·25-s + 1.35·26-s + 1.85·27-s − 0.478·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.181·3-s + 0.5·4-s + 1.54·5-s − 0.128·6-s − 0.180·7-s + 0.353·8-s − 0.967·9-s + 1.09·10-s + 0.695·11-s − 0.0907·12-s + 0.375·13-s − 0.127·14-s − 0.281·15-s + 0.250·16-s − 0.155·17-s − 0.683·18-s + 0.359·19-s + 0.774·20-s + 0.0328·21-s + 0.491·22-s + 0.479·23-s − 0.0641·24-s + 1.40·25-s + 0.265·26-s + 0.356·27-s − 0.0903·28-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
| \(L(1)\) |
\(\approx\) |
\(3.905206005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.905206005\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.314T + 3T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 0.478T + 7T^{2} \) |
| 11 | \( 1 - 2.30T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 + 0.642T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 - 2.29T + 23T^{2} \) |
| 29 | \( 1 - 0.990T + 29T^{2} \) |
| 31 | \( 1 + 2.42T + 31T^{2} \) |
| 37 | \( 1 - 1.54T + 37T^{2} \) |
| 41 | \( 1 - 0.666T + 41T^{2} \) |
| 43 | \( 1 - 6.84T + 43T^{2} \) |
| 47 | \( 1 - 8.00T + 47T^{2} \) |
| 53 | \( 1 - 3.07T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 1.96T + 67T^{2} \) |
| 71 | \( 1 + 5.97T + 71T^{2} \) |
| 73 | \( 1 - 2.10T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 9.39T + 83T^{2} \) |
| 89 | \( 1 - 6.44T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
| 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
−8.632361716443110327486575043931, −7.52376928296750752179073946031, −6.59266798222658030457432005547, −6.12609181418998649949909202914, −5.56532467057668332689929829535, −4.90422779934617220084968829655, −3.81695887712029605071954792866, −2.89503870116900607012124829887, −2.15656007214653636854488288302, −1.08837017968464891419560507516,
1.08837017968464891419560507516, 2.15656007214653636854488288302, 2.89503870116900607012124829887, 3.81695887712029605071954792866, 4.90422779934617220084968829655, 5.56532467057668332689929829535, 6.12609181418998649949909202914, 6.59266798222658030457432005547, 7.52376928296750752179073946031, 8.632361716443110327486575043931
