| L(s) = 1 | + 2-s + 4-s − 3·5-s + 8-s − 3·10-s − 3·11-s + 4·13-s + 16-s − 6·17-s + 7·19-s − 3·20-s − 3·22-s + 3·23-s + 4·25-s + 4·26-s − 5·31-s + 32-s − 6·34-s − 7·37-s + 7·38-s − 3·40-s − 9·41-s − 10·43-s − 3·44-s + 3·46-s + 6·47-s + 4·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s − 0.904·11-s + 1.10·13-s + 1/4·16-s − 1.45·17-s + 1.60·19-s − 0.670·20-s − 0.639·22-s + 0.625·23-s + 4/5·25-s + 0.784·26-s − 0.898·31-s + 0.176·32-s − 1.02·34-s − 1.15·37-s + 1.13·38-s − 0.474·40-s − 1.40·41-s − 1.52·43-s − 0.452·44-s + 0.442·46-s + 0.875·47-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−8.411878978422576008118242977401, −7.54093826265732890551104966146, −7.07803356069366613643720682431, −6.12585114739427978407313474137, −5.13858233863725778435685317082, −4.55558448712373822916009028976, −3.51027276651897791446234749457, −3.11258133172613112703068504748, −1.63127189692849427124845257150, 0,
1.63127189692849427124845257150, 3.11258133172613112703068504748, 3.51027276651897791446234749457, 4.55558448712373822916009028976, 5.13858233863725778435685317082, 6.12585114739427978407313474137, 7.07803356069366613643720682431, 7.54093826265732890551104966146, 8.411878978422576008118242977401
