| L(s) = 1 | − 2-s + 1.44·3-s + 4-s + 0.168·5-s − 1.44·6-s − 1.52·7-s − 8-s − 0.917·9-s − 0.168·10-s + 11-s + 1.44·12-s − 3.01·13-s + 1.52·14-s + 0.243·15-s + 16-s + 3.63·17-s + 0.917·18-s + 0.168·20-s − 2.20·21-s − 22-s − 4.60·23-s − 1.44·24-s − 4.97·25-s + 3.01·26-s − 5.65·27-s − 1.52·28-s − 1.25·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.833·3-s + 0.5·4-s + 0.0753·5-s − 0.589·6-s − 0.576·7-s − 0.353·8-s − 0.305·9-s − 0.0533·10-s + 0.301·11-s + 0.416·12-s − 0.837·13-s + 0.407·14-s + 0.0628·15-s + 0.250·16-s + 0.881·17-s + 0.216·18-s + 0.0376·20-s − 0.480·21-s − 0.213·22-s − 0.959·23-s − 0.294·24-s − 0.994·25-s + 0.591·26-s − 1.08·27-s − 0.288·28-s − 0.232·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.408256382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.408256382\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 - 0.168T + 5T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 13 | \( 1 + 3.01T + 13T^{2} \) |
| 17 | \( 1 - 3.63T + 17T^{2} \) |
| 23 | \( 1 + 4.60T + 23T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 - 3.10T + 31T^{2} \) |
| 37 | \( 1 - 8.91T + 37T^{2} \) |
| 41 | \( 1 - 3.30T + 41T^{2} \) |
| 43 | \( 1 + 5.92T + 43T^{2} \) |
| 47 | \( 1 - 7.55T + 47T^{2} \) |
| 53 | \( 1 + 7.04T + 53T^{2} \) |
| 59 | \( 1 + 7.74T + 59T^{2} \) |
| 61 | \( 1 - 7.64T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 1.69T + 79T^{2} \) |
| 83 | \( 1 - 6.75T + 83T^{2} \) |
| 89 | \( 1 + 6.28T + 89T^{2} \) |
| 97 | \( 1 + 1.21T + 97T^{2} \) |
| 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
−7.954408810341424445621896252770, −7.46100300272184964459573731250, −6.50006233404098163833512225247, −5.97907840571373557831623682868, −5.14226086372099257196292116159, −4.00276698109284991817750543807, −3.35873734708498967621596978086, −2.55093639180790996375764557277, −1.92925527171664797568474443232, −0.60172693476895489463793242524,
0.60172693476895489463793242524, 1.92925527171664797568474443232, 2.55093639180790996375764557277, 3.35873734708498967621596978086, 4.00276698109284991817750543807, 5.14226086372099257196292116159, 5.97907840571373557831623682868, 6.50006233404098163833512225247, 7.46100300272184964459573731250, 7.954408810341424445621896252770
