| L(s) = 1 | − 5-s + 7-s − 13-s − 8·17-s − 5·19-s − 4·25-s + 5·29-s − 6·31-s − 35-s − 11·37-s − 8·43-s − 7·47-s + 49-s − 2·53-s − 7·59-s + 2·61-s + 65-s − 7·67-s − 7·73-s − 2·79-s − 12·83-s + 8·85-s − 2·89-s − 91-s + 5·95-s − 8·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.277·13-s − 1.94·17-s − 1.14·19-s − 4/5·25-s + 0.928·29-s − 1.07·31-s − 0.169·35-s − 1.80·37-s − 1.21·43-s − 1.02·47-s + 1/7·49-s − 0.274·53-s − 0.911·59-s + 0.256·61-s + 0.124·65-s − 0.855·67-s − 0.819·73-s − 0.225·79-s − 1.31·83-s + 0.867·85-s − 0.211·89-s − 0.104·91-s + 0.512·95-s − 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.99140250951945, −14.28166635704180, −13.77345474560792, −13.28653376053629, −12.79696831365142, −12.26050812728450, −11.72365162572515, −11.22847505165528, −10.83245459009197, −10.28690755053387, −9.737617319473480, −8.957840994553419, −8.591451776912765, −8.256586789609768, −7.455066529463297, −6.979055065480626, −6.492794652271243, −5.920279847112559, −5.080886417384186, −4.636019720726400, −4.140822960595635, −3.501680165823352, −2.727983685099048, −1.967464471674923, −1.561536782890750, 0, 0,
1.561536782890750, 1.967464471674923, 2.727983685099048, 3.501680165823352, 4.140822960595635, 4.636019720726400, 5.080886417384186, 5.920279847112559, 6.492794652271243, 6.979055065480626, 7.455066529463297, 8.256586789609768, 8.591451776912765, 8.957840994553419, 9.737617319473480, 10.28690755053387, 10.83245459009197, 11.22847505165528, 11.72365162572515, 12.26050812728450, 12.79696831365142, 13.28653376053629, 13.77345474560792, 14.28166635704180, 14.99140250951945
