| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 4·11-s + 2·13-s + 15-s + 17-s + 4·19-s + 4·21-s + 25-s + 27-s − 2·29-s − 4·33-s + 4·35-s − 2·37-s + 2·39-s + 6·41-s + 8·43-s + 45-s − 8·47-s + 9·49-s + 51-s + 6·53-s − 4·55-s + 4·57-s + 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.242·17-s + 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.676·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s + 0.140·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.238639287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.238639287\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.470770728506762982214955189573, −7.61120582262229803259959075798, −7.40482834259110305072425292563, −6.07222000001737998766751687749, −5.34826462308760317540143328088, −4.79115526258017019169544729755, −3.82256359171002894408181453537, −2.79516385947658142038022464125, −2.01160533963847894191314784515, −1.07784062898596240556871145354,
1.07784062898596240556871145354, 2.01160533963847894191314784515, 2.79516385947658142038022464125, 3.82256359171002894408181453537, 4.79115526258017019169544729755, 5.34826462308760317540143328088, 6.07222000001737998766751687749, 7.40482834259110305072425292563, 7.61120582262229803259959075798, 8.470770728506762982214955189573
