| L(s) = 1 | + 2·5-s − 2·7-s − 11-s − 4·13-s − 17-s − 2·19-s − 2·23-s − 25-s + 2·29-s − 4·31-s − 4·35-s − 6·37-s − 6·41-s − 2·43-s − 3·49-s − 12·53-s − 2·55-s − 14·59-s − 6·61-s − 8·65-s + 4·67-s + 2·71-s − 8·73-s + 2·77-s − 2·79-s − 12·83-s − 2·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.301·11-s − 1.10·13-s − 0.242·17-s − 0.458·19-s − 0.417·23-s − 1/5·25-s + 0.371·29-s − 0.718·31-s − 0.676·35-s − 0.986·37-s − 0.937·41-s − 0.304·43-s − 3/7·49-s − 1.64·53-s − 0.269·55-s − 1.82·59-s − 0.768·61-s − 0.992·65-s + 0.488·67-s + 0.237·71-s − 0.936·73-s + 0.227·77-s − 0.225·79-s − 1.31·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.16617369913721, −13.67999642258289, −13.15433328180134, −12.75282812062731, −12.31232544797907, −11.87558090457441, −11.16196986172690, −10.61349874656999, −10.19453051174487, −9.680303045705887, −9.453054758872813, −8.829834958352708, −8.224360315222011, −7.683057218579989, −7.051330246495157, −6.625782230898347, −6.085366877071747, −5.632958639575192, −4.974045732896608, −4.573365602846377, −3.776512261723868, −3.086798425251475, −2.663795928642901, −1.874180601371369, −1.515821626852520, 0, 0,
1.515821626852520, 1.874180601371369, 2.663795928642901, 3.086798425251475, 3.776512261723868, 4.573365602846377, 4.974045732896608, 5.632958639575192, 6.085366877071747, 6.625782230898347, 7.051330246495157, 7.683057218579989, 8.224360315222011, 8.829834958352708, 9.453054758872813, 9.680303045705887, 10.19453051174487, 10.61349874656999, 11.16196986172690, 11.87558090457441, 12.31232544797907, 12.75282812062731, 13.15433328180134, 13.67999642258289, 14.16617369913721
