| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s − 4·11-s + 2·13-s − 2·15-s − 6·17-s − 4·19-s + 21-s − 25-s − 27-s + 2·29-s + 4·33-s − 2·35-s − 6·37-s − 2·39-s + 2·41-s + 4·43-s + 2·45-s + 49-s + 6·51-s − 6·53-s − 8·55-s + 4·57-s − 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-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 + 0.218·21-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s − 0.338·35-s − 0.986·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 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
−9.238117136499894867345980510658, −8.535675361637489516400731752225, −7.48171223884197946068743795147, −6.47604188152629865155494776849, −6.00140300914464627775014931904, −5.08548839023881336508605977410, −4.19315915887555872024919481174, −2.78781035361267366934893473837, −1.80984559120091606425042866771, 0,
1.80984559120091606425042866771, 2.78781035361267366934893473837, 4.19315915887555872024919481174, 5.08548839023881336508605977410, 6.00140300914464627775014931904, 6.47604188152629865155494776849, 7.48171223884197946068743795147, 8.535675361637489516400731752225, 9.238117136499894867345980510658
