| L(s) = 1 | − 2·5-s − 7-s − 3·9-s + 2·13-s + 17-s + 4·19-s − 25-s + 6·29-s + 2·35-s + 6·37-s − 6·41-s − 12·43-s + 6·45-s − 8·47-s + 49-s + 2·53-s + 4·59-s − 2·61-s + 3·63-s − 4·65-s + 12·67-s + 2·73-s + 8·79-s + 9·81-s + 12·83-s − 2·85-s + 10·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 9-s + 0.554·13-s + 0.242·17-s + 0.917·19-s − 1/5·25-s + 1.11·29-s + 0.338·35-s + 0.986·37-s − 0.937·41-s − 1.82·43-s + 0.894·45-s − 1.16·47-s + 1/7·49-s + 0.274·53-s + 0.520·59-s − 0.256·61-s + 0.377·63-s − 0.496·65-s + 1.46·67-s + 0.234·73-s + 0.900·79-s + 81-s + 1.31·83-s − 0.216·85-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.71282024817693675745013733529, −6.73763985186732377586137626470, −6.26990474233050369942971608193, −5.35539917644036382158361461516, −4.77682145642589218436996001225, −3.63026567116244883219000059570, −3.36366542167825731325973439863, −2.40395727895840686691357326236, −1.08252162774259574666896222029, 0,
1.08252162774259574666896222029, 2.40395727895840686691357326236, 3.36366542167825731325973439863, 3.63026567116244883219000059570, 4.77682145642589218436996001225, 5.35539917644036382158361461516, 6.26990474233050369942971608193, 6.73763985186732377586137626470, 7.71282024817693675745013733529
