| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s + 4·11-s − 2·13-s − 15-s − 2·17-s + 4·21-s + 8·23-s + 25-s − 27-s + 6·29-s − 4·33-s − 4·35-s + 37-s + 2·39-s + 2·41-s − 8·43-s + 45-s − 8·47-s + 9·49-s + 2·51-s − 6·53-s + 4·55-s + 4·59-s + 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.872·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.696·33-s − 0.676·35-s + 0.164·37-s + 0.320·39-s + 0.312·41-s − 1.21·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s + 0.280·51-s − 0.824·53-s + 0.539·55-s + 0.520·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.455912905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455912905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.01096028661636, −15.31507930582395, −14.71070323153674, −14.23989315047316, −13.42683546466116, −12.93586541035430, −12.72364531564235, −11.81171875129941, −11.56038736541721, −10.75597290922151, −10.00169870041412, −9.835691106975973, −9.018334987617599, −8.784129063767111, −7.653127686116998, −6.828721198391801, −6.578364367218245, −6.208538390619290, −5.226859776394247, −4.755069647741118, −3.850722258525411, −3.197077176740674, −2.499168714960221, −1.418274929225872, −0.5528387412103392,
0.5528387412103392, 1.418274929225872, 2.499168714960221, 3.197077176740674, 3.850722258525411, 4.755069647741118, 5.226859776394247, 6.208538390619290, 6.578364367218245, 6.828721198391801, 7.653127686116998, 8.784129063767111, 9.018334987617599, 9.835691106975973, 10.00169870041412, 10.75597290922151, 11.56038736541721, 11.81171875129941, 12.72364531564235, 12.93586541035430, 13.42683546466116, 14.23989315047316, 14.71070323153674, 15.31507930582395, 16.01096028661636
