L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s − 2·13-s + 14-s + 16-s + 6·17-s − 4·19-s − 20-s − 22-s + 25-s + 2·26-s − 28-s + 2·29-s − 8·31-s − 32-s − 6·34-s + 35-s + 6·37-s + 4·38-s + 40-s + 6·41-s − 4·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.223·20-s − 0.213·22-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.371·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.169·35-s + 0.986·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 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 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.64623491255279602211132094566, −7.10168387856966897474617584387, −6.31571712911772235090830112272, −5.65289343212575187816347972699, −4.73903104675503567842529946487, −3.82220536900393063633329498198, −3.12573459239352076111884130123, −2.19521899758026488577581567384, −1.12337533148093317196533160639, 0,
1.12337533148093317196533160639, 2.19521899758026488577581567384, 3.12573459239352076111884130123, 3.82220536900393063633329498198, 4.73903104675503567842529946487, 5.65289343212575187816347972699, 6.31571712911772235090830112272, 7.10168387856966897474617584387, 7.64623491255279602211132094566