| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 11-s − 2·13-s − 14-s + 16-s − 2·17-s − 4·19-s − 20-s − 22-s − 8·23-s + 25-s + 2·26-s + 28-s − 6·29-s + 8·31-s − 32-s + 2·34-s − 35-s − 2·37-s + 4·38-s + 40-s − 2·41-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 − 0.485·17-s − 0.917·19-s − 0.223·20-s − 0.213·22-s − 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 0.169·35-s − 0.328·37-s + 0.648·38-s + 0.158·40-s − 0.312·41-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
| \(L(1)\) |
\(\approx\) |
\(0.9075788965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9075788965\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−7.973713173933110947926764864448, −7.45935873328665285211014998706, −6.62124918534133464749967251506, −6.08664865510948583917892553092, −5.10046287983341572633808332872, −4.28569018834161615989680894313, −3.62964123203857338701229904242, −2.43690865935129462013858630439, −1.83614978588270814499130694422, −0.52531453803820092568863048296,
0.52531453803820092568863048296, 1.83614978588270814499130694422, 2.43690865935129462013858630439, 3.62964123203857338701229904242, 4.28569018834161615989680894313, 5.10046287983341572633808332872, 6.08664865510948583917892553092, 6.62124918534133464749967251506, 7.45935873328665285211014998706, 7.973713173933110947926764864448
