| L(s) = 1 | − 5-s + 1.64·7-s + 0.354·11-s + 3.64·13-s − 19-s − 2·23-s + 25-s − 9.64·29-s − 2·31-s − 1.64·35-s − 6.93·37-s − 1.64·41-s − 4.93·43-s − 6·47-s − 4.29·49-s − 4·53-s − 0.354·55-s − 3.29·59-s + 11.2·61-s − 3.64·65-s + 4·67-s + 3.29·71-s − 2·73-s + 0.583·77-s − 8·79-s + 15.8·83-s + 12.2·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.622·7-s + 0.106·11-s + 1.01·13-s − 0.229·19-s − 0.417·23-s + 0.200·25-s − 1.79·29-s − 0.359·31-s − 0.278·35-s − 1.14·37-s − 0.257·41-s − 0.752·43-s − 0.875·47-s − 0.613·49-s − 0.549·53-s − 0.0477·55-s − 0.428·59-s + 1.44·61-s − 0.452·65-s + 0.488·67-s + 0.390·71-s − 0.234·73-s + 0.0664·77-s − 0.900·79-s + 1.74·83-s + 1.29·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 - 0.354T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 9.64T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 6.93T + 37T^{2} \) |
| 41 | \( 1 + 1.64T + 41T^{2} \) |
| 43 | \( 1 + 4.93T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
| 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
−7.77468048188027415214464814557, −6.91691587776728930274134531991, −6.27268838839961377524808799520, −5.41462947043037261586227133427, −4.81024182804187888381928171850, −3.79469421117479218001360529921, −3.46401341854821465178807257301, −2.11741318999224700598560461554, −1.38662167905908509697987813700, 0,
1.38662167905908509697987813700, 2.11741318999224700598560461554, 3.46401341854821465178807257301, 3.79469421117479218001360529921, 4.81024182804187888381928171850, 5.41462947043037261586227133427, 6.27268838839961377524808799520, 6.91691587776728930274134531991, 7.77468048188027415214464814557
