| L(s) = 1 | − 1.28·2-s − 0.829·3-s − 0.348·4-s − 5-s + 1.06·6-s − 7-s + 3.01·8-s − 2.31·9-s + 1.28·10-s − 2.11·11-s + 0.288·12-s + 3.07·13-s + 1.28·14-s + 0.829·15-s − 3.18·16-s − 6.86·17-s + 2.97·18-s − 3.15·19-s + 0.348·20-s + 0.829·21-s + 2.72·22-s − 0.512·23-s − 2.50·24-s + 25-s − 3.94·26-s + 4.40·27-s + 0.348·28-s + ⋯ |
| L(s) = 1 | − 0.908·2-s − 0.479·3-s − 0.174·4-s − 0.447·5-s + 0.435·6-s − 0.377·7-s + 1.06·8-s − 0.770·9-s + 0.406·10-s − 0.639·11-s + 0.0833·12-s + 0.851·13-s + 0.343·14-s + 0.214·15-s − 0.795·16-s − 1.66·17-s + 0.700·18-s − 0.722·19-s + 0.0778·20-s + 0.181·21-s + 0.580·22-s − 0.106·23-s − 0.511·24-s + 0.200·25-s − 0.774·26-s + 0.848·27-s + 0.0657·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08637762386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08637762386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 1.28T + 2T^{2} \) |
| 3 | \( 1 + 0.829T + 3T^{2} \) |
| 11 | \( 1 + 2.11T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 17 | \( 1 + 6.86T + 17T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 23 | \( 1 + 0.512T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 - 3.72T + 31T^{2} \) |
| 37 | \( 1 + 5.59T + 37T^{2} \) |
| 41 | \( 1 + 7.64T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 5.76T + 53T^{2} \) |
| 59 | \( 1 + 0.788T + 59T^{2} \) |
| 61 | \( 1 - 5.99T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 9.76T + 73T^{2} \) |
| 79 | \( 1 + 9.51T + 79T^{2} \) |
| 83 | \( 1 + 2.71T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 0.759T + 97T^{2} \) |
| 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
−8.236922312365316908496618110035, −7.09438770790158255247054189169, −6.62950650944798661542906051064, −5.87146025717726194044652279788, −4.94173287491196179779387575056, −4.43369621024861923655113599083, −3.49907114919209449975031577909, −2.56128391355565289642721742626, −1.48220567321819172890633832696, −0.17540525204923842794598338497,
0.17540525204923842794598338497, 1.48220567321819172890633832696, 2.56128391355565289642721742626, 3.49907114919209449975031577909, 4.43369621024861923655113599083, 4.94173287491196179779387575056, 5.87146025717726194044652279788, 6.62950650944798661542906051064, 7.09438770790158255247054189169, 8.236922312365316908496618110035
