| L(s) = 1 | − 3-s + 2·5-s − 2·7-s + 9-s + 4·11-s + 13-s − 2·15-s + 17-s − 8·19-s + 2·21-s − 25-s − 27-s + 2·29-s − 6·31-s − 4·33-s − 4·35-s − 4·37-s − 39-s + 2·41-s − 4·43-s + 2·45-s − 6·47-s − 3·49-s − 51-s + 6·53-s + 8·55-s + 8·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.516·15-s + 0.242·17-s − 1.83·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.07·31-s − 0.696·33-s − 0.676·35-s − 0.657·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s − 0.875·47-s − 3/7·49-s − 0.140·51-s + 0.824·53-s + 1.07·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.79242811797529697804374435123, −6.65931863764501138275153569426, −6.48024687715223345674067083094, −5.86981020265139472105259859742, −5.00172333144528835545765914493, −4.08965967187448867946783201580, −3.40597410087395702389954307665, −2.17181940123828302054512728548, −1.41553242104059504572565866235, 0,
1.41553242104059504572565866235, 2.17181940123828302054512728548, 3.40597410087395702389954307665, 4.08965967187448867946783201580, 5.00172333144528835545765914493, 5.86981020265139472105259859742, 6.48024687715223345674067083094, 6.65931863764501138275153569426, 7.79242811797529697804374435123
