| L(s) = 1 | − 3-s − 5-s + 2·7-s − 2·9-s + 11-s − 4·13-s + 15-s − 2·17-s − 2·21-s + 23-s − 4·25-s + 5·27-s − 7·31-s − 33-s − 2·35-s − 3·37-s + 4·39-s − 8·41-s − 6·43-s + 2·45-s − 8·47-s − 3·49-s + 2·51-s + 6·53-s − 55-s + 5·59-s − 12·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s − 2/3·9-s + 0.301·11-s − 1.10·13-s + 0.258·15-s − 0.485·17-s − 0.436·21-s + 0.208·23-s − 4/5·25-s + 0.962·27-s − 1.25·31-s − 0.174·33-s − 0.338·35-s − 0.493·37-s + 0.640·39-s − 1.24·41-s − 0.914·43-s + 0.298·45-s − 1.16·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 0.134·55-s + 0.650·59-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.14466147519938939509327606244, −9.092146666164135933217503972545, −8.248674172646585313129242503450, −7.39701457500400876162065539217, −6.46350440936163119026889680667, −5.32992026591659065835464612635, −4.69659420732828154227208765880, −3.40579821695092674516778014630, −1.94547887078347189383086322028, 0,
1.94547887078347189383086322028, 3.40579821695092674516778014630, 4.69659420732828154227208765880, 5.32992026591659065835464612635, 6.46350440936163119026889680667, 7.39701457500400876162065539217, 8.248674172646585313129242503450, 9.092146666164135933217503972545, 10.14466147519938939509327606244
