| L(s) = 1 | + 2-s − 2.47·3-s + 4-s − 5-s − 2.47·6-s + 7-s + 8-s + 3.11·9-s − 10-s + 1.78·11-s − 2.47·12-s + 6.30·13-s + 14-s + 2.47·15-s + 16-s − 0.389·17-s + 3.11·18-s + 5.77·19-s − 20-s − 2.47·21-s + 1.78·22-s − 6.21·23-s − 2.47·24-s + 25-s + 6.30·26-s − 0.275·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.42·3-s + 0.5·4-s − 0.447·5-s − 1.00·6-s + 0.377·7-s + 0.353·8-s + 1.03·9-s − 0.316·10-s + 0.536·11-s − 0.713·12-s + 1.74·13-s + 0.267·14-s + 0.638·15-s + 0.250·16-s − 0.0944·17-s + 0.733·18-s + 1.32·19-s − 0.223·20-s − 0.539·21-s + 0.379·22-s − 1.29·23-s − 0.504·24-s + 0.200·25-s + 1.23·26-s − 0.0531·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.067849008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.067849008\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| good | 3 | \( 1 + 2.47T + 3T^{2} \) |
| 11 | \( 1 - 1.78T + 11T^{2} \) |
| 13 | \( 1 - 6.30T + 13T^{2} \) |
| 17 | \( 1 + 0.389T + 17T^{2} \) |
| 19 | \( 1 - 5.77T + 19T^{2} \) |
| 23 | \( 1 + 6.21T + 23T^{2} \) |
| 29 | \( 1 - 4.10T + 29T^{2} \) |
| 31 | \( 1 - 1.11T + 31T^{2} \) |
| 37 | \( 1 - 0.848T + 37T^{2} \) |
| 41 | \( 1 + 2.99T + 41T^{2} \) |
| 43 | \( 1 + 3.99T + 43T^{2} \) |
| 47 | \( 1 + 0.176T + 47T^{2} \) |
| 53 | \( 1 + 2.69T + 53T^{2} \) |
| 59 | \( 1 + 5.64T + 59T^{2} \) |
| 61 | \( 1 - 4.43T + 61T^{2} \) |
| 67 | \( 1 + 6.79T + 67T^{2} \) |
| 73 | \( 1 - 3.30T + 73T^{2} \) |
| 79 | \( 1 + 4.21T + 79T^{2} \) |
| 83 | \( 1 - 7.75T + 83T^{2} \) |
| 89 | \( 1 - 5.36T + 89T^{2} \) |
| 97 | \( 1 - 7.72T + 97T^{2} \) |
| 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
−8.107207879941651985535095365640, −7.32462052858663305466629119455, −6.39135571188488776195816498132, −6.14645993329677284467031579311, −5.34086694693791935735996275137, −4.66163812204223219770950574329, −3.91234086306054972822762469536, −3.19567045355812176803583701827, −1.65979276000868938968171328799, −0.817595327698809316378360027035,
0.817595327698809316378360027035, 1.65979276000868938968171328799, 3.19567045355812176803583701827, 3.91234086306054972822762469536, 4.66163812204223219770950574329, 5.34086694693791935735996275137, 6.14645993329677284467031579311, 6.39135571188488776195816498132, 7.32462052858663305466629119455, 8.107207879941651985535095365640
