L(s) = 1 | + 1.30·2-s + 2.43·3-s − 0.285·4-s + 3.19·6-s + 7-s − 2.99·8-s + 2.95·9-s + 0.628·11-s − 0.697·12-s − 3.82·13-s + 1.30·14-s − 3.34·16-s + 5.22·17-s + 3.86·18-s + 6.31·19-s + 2.43·21-s + 0.823·22-s + 23-s − 7.30·24-s − 5.00·26-s − 0.115·27-s − 0.285·28-s + 3.73·29-s + 9.35·31-s + 1.60·32-s + 1.53·33-s + 6.83·34-s + ⋯ |
L(s) = 1 | + 0.925·2-s + 1.40·3-s − 0.142·4-s + 1.30·6-s + 0.377·7-s − 1.05·8-s + 0.984·9-s + 0.189·11-s − 0.201·12-s − 1.05·13-s + 0.349·14-s − 0.836·16-s + 1.26·17-s + 0.911·18-s + 1.44·19-s + 0.532·21-s + 0.175·22-s + 0.208·23-s − 1.49·24-s − 0.981·26-s − 0.0221·27-s − 0.0540·28-s + 0.694·29-s + 1.68·31-s + 0.283·32-s + 0.267·33-s + 1.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.778141589\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.778141589\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 3 | \( 1 - 2.43T + 3T^{2} \) |
| 11 | \( 1 - 0.628T + 11T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 6.31T + 19T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 - 9.35T + 31T^{2} \) |
| 37 | \( 1 + 7.74T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 4.67T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 8.98T + 53T^{2} \) |
| 59 | \( 1 + 4.34T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 + 6.33T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 7.44T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 0.664T + 83T^{2} \) |
| 89 | \( 1 + 9.82T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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
−8.510600379848186181000720494553, −7.64997165939311862685287793424, −7.24410414795897929234943151026, −6.01575548211013718336036916899, −5.26191079752578407700939176187, −4.57786962294286879874165523650, −3.76084230604956058881817975899, −3.00068003187579669018070258374, −2.51896659148113073338860891831, −1.08045526157825141781243093686,
1.08045526157825141781243093686, 2.51896659148113073338860891831, 3.00068003187579669018070258374, 3.76084230604956058881817975899, 4.57786962294286879874165523650, 5.26191079752578407700939176187, 6.01575548211013718336036916899, 7.24410414795897929234943151026, 7.64997165939311862685287793424, 8.510600379848186181000720494553