| L(s) = 1 | + 2.52·2-s + 3-s + 4.38·4-s − 0.629·5-s + 2.52·6-s − 7-s + 6.03·8-s + 9-s − 1.59·10-s + 3.78·11-s + 4.38·12-s + 4.23·13-s − 2.52·14-s − 0.629·15-s + 6.48·16-s − 3.56·17-s + 2.52·18-s + 4.76·19-s − 2.76·20-s − 21-s + 9.57·22-s + 5.38·23-s + 6.03·24-s − 4.60·25-s + 10.6·26-s + 27-s − 4.38·28-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 0.577·3-s + 2.19·4-s − 0.281·5-s + 1.03·6-s − 0.377·7-s + 2.13·8-s + 0.333·9-s − 0.502·10-s + 1.14·11-s + 1.26·12-s + 1.17·13-s − 0.675·14-s − 0.162·15-s + 1.62·16-s − 0.865·17-s + 0.595·18-s + 1.09·19-s − 0.617·20-s − 0.218·21-s + 2.04·22-s + 1.12·23-s + 1.23·24-s − 0.920·25-s + 2.09·26-s + 0.192·27-s − 0.829·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.793528847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.793528847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 5 | \( 1 + 0.629T + 5T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 - 4.76T + 19T^{2} \) |
| 23 | \( 1 - 5.38T + 23T^{2} \) |
| 29 | \( 1 - 0.882T + 29T^{2} \) |
| 31 | \( 1 + 0.516T + 31T^{2} \) |
| 37 | \( 1 + 9.44T + 37T^{2} \) |
| 41 | \( 1 - 6.85T + 41T^{2} \) |
| 43 | \( 1 + 1.17T + 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 + 3.19T + 53T^{2} \) |
| 59 | \( 1 + 6.46T + 59T^{2} \) |
| 61 | \( 1 - 2.38T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 7.26T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 6.93T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 1.62T + 97T^{2} \) |
| 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
−7.39387808990196948698933590495, −7.01430816303626550862074244260, −6.28287100644351551362545859567, −5.75117627648322222929724487219, −4.85160285194801824970495989144, −4.13220305537479970232599950669, −3.58246374432415978731053784195, −3.12540377606743304833072098680, −2.12067441189268924809312771466, −1.19035873402339681626107268629,
1.19035873402339681626107268629, 2.12067441189268924809312771466, 3.12540377606743304833072098680, 3.58246374432415978731053784195, 4.13220305537479970232599950669, 4.85160285194801824970495989144, 5.75117627648322222929724487219, 6.28287100644351551362545859567, 7.01430816303626550862074244260, 7.39387808990196948698933590495
