| L(s) = 1 | + 2-s − 3-s − 4-s − 2·5-s − 6-s − 7-s − 3·8-s + 9-s − 2·10-s − 4·11-s + 12-s − 2·13-s − 14-s + 2·15-s − 16-s + 6·17-s + 18-s − 4·19-s + 2·20-s + 21-s − 4·22-s + 3·24-s − 25-s − 2·26-s − 27-s + 28-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.852·22-s + 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 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
−15.92322179383795, −15.59320783890561, −14.90096617822985, −14.61579613833399, −13.76908760732145, −13.31771988828965, −12.68470138248462, −12.24207991201229, −12.01358710163982, −11.18406361760242, −10.54909383603323, −9.984210824508175, −9.525567239363557, −8.548328603682253, −8.135249988088523, −7.482924638457681, −6.855407505271171, −6.047184991050379, −5.331274886661726, −5.140834487625315, −4.207552424639129, −3.750076612974927, −3.052790402945715, −2.223180034932175, −0.7375687472295627, 0,
0.7375687472295627, 2.223180034932175, 3.052790402945715, 3.750076612974927, 4.207552424639129, 5.140834487625315, 5.331274886661726, 6.047184991050379, 6.855407505271171, 7.482924638457681, 8.135249988088523, 8.548328603682253, 9.525567239363557, 9.984210824508175, 10.54909383603323, 11.18406361760242, 12.01358710163982, 12.24207991201229, 12.68470138248462, 13.31771988828965, 13.76908760732145, 14.61579613833399, 14.90096617822985, 15.59320783890561, 15.92322179383795
