| L(s) = 1 | − 5-s − 5·11-s + 13-s − 3·17-s + 8·19-s + 3·23-s + 25-s + 2·29-s + 10·31-s − 3·37-s + 3·41-s − 10·43-s − 6·47-s − 5·53-s + 5·55-s + 4·59-s + 7·61-s − 65-s + 12·67-s − 5·71-s − 2·73-s − 13·79-s − 12·83-s + 3·85-s − 7·89-s − 8·95-s + 5·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.50·11-s + 0.277·13-s − 0.727·17-s + 1.83·19-s + 0.625·23-s + 1/5·25-s + 0.371·29-s + 1.79·31-s − 0.493·37-s + 0.468·41-s − 1.52·43-s − 0.875·47-s − 0.686·53-s + 0.674·55-s + 0.520·59-s + 0.896·61-s − 0.124·65-s + 1.46·67-s − 0.593·71-s − 0.234·73-s − 1.46·79-s − 1.31·83-s + 0.325·85-s − 0.741·89-s − 0.820·95-s + 0.507·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.810585226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.810585226\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.09387389263696, −12.51920342255291, −11.86743370396420, −11.54199747368957, −11.19010296403434, −10.61169637591013, −10.02272136699517, −9.872197215765907, −9.211423232656188, −8.451410606559885, −8.294903474091110, −7.835536648134775, −7.153402831315324, −6.895592843029683, −6.294273817482296, −5.451381026119402, −5.334482474896850, −4.622362888139189, −4.320935018753368, −3.292543072329476, −3.124493704722258, −2.577941985418985, −1.800497776153595, −1.038320763710582, −0.4201968669095359,
0.4201968669095359, 1.038320763710582, 1.800497776153595, 2.577941985418985, 3.124493704722258, 3.292543072329476, 4.320935018753368, 4.622362888139189, 5.334482474896850, 5.451381026119402, 6.294273817482296, 6.895592843029683, 7.153402831315324, 7.835536648134775, 8.294903474091110, 8.451410606559885, 9.211423232656188, 9.872197215765907, 10.02272136699517, 10.61169637591013, 11.19010296403434, 11.54199747368957, 11.86743370396420, 12.51920342255291, 13.09387389263696
