| L(s) = 1 | + 3-s − 2·5-s + 9-s − 4·11-s − 2·13-s − 2·15-s + 6·17-s + 4·19-s − 25-s + 27-s + 2·29-s − 4·33-s − 6·37-s − 2·39-s − 2·41-s + 4·43-s − 2·45-s + 6·51-s − 6·53-s + 8·55-s + 4·57-s + 12·59-s − 2·61-s + 4·65-s − 4·67-s + 6·73-s − 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.696·33-s − 0.986·37-s − 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.298·45-s + 0.840·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s + 1.56·59-s − 0.256·61-s + 0.496·65-s − 0.488·67-s + 0.702·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.41162979563327424538391535036, −7.08463284614449031614302730659, −5.84366683169447599172191319438, −5.25233061805071063226919136648, −4.56778712711941505527081631821, −3.63417394911967134530507000276, −3.13295449004887072368969635069, −2.36822825472434546359232655114, −1.19651242282215311581594916621, 0,
1.19651242282215311581594916621, 2.36822825472434546359232655114, 3.13295449004887072368969635069, 3.63417394911967134530507000276, 4.56778712711941505527081631821, 5.25233061805071063226919136648, 5.84366683169447599172191319438, 7.08463284614449031614302730659, 7.41162979563327424538391535036
