| L(s) = 1 | − 2·5-s − 3·9-s + 6·13-s − 2·17-s − 8·19-s − 25-s − 2·29-s + 8·31-s + 2·37-s + 6·41-s + 4·43-s + 6·45-s − 6·53-s − 4·59-s − 6·61-s − 12·65-s + 4·67-s + 8·71-s + 6·73-s + 8·79-s + 9·81-s + 4·85-s + 6·89-s + 16·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s + 1.66·13-s − 0.485·17-s − 1.83·19-s − 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 0.894·45-s − 0.824·53-s − 0.520·59-s − 0.768·61-s − 1.48·65-s + 0.488·67-s + 0.949·71-s + 0.702·73-s + 0.900·79-s + 81-s + 0.433·85-s + 0.635·89-s + 1.64·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 362992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 362992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.389104313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389104313\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 463 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.43853008318337, −12.09830317416286, −11.44018880510507, −11.17220742043455, −10.82476118035061, −10.55513614447631, −9.710526821649313, −9.224893428565536, −8.747734111669649, −8.391702617364969, −7.996045490445856, −7.724622126299574, −6.841965700006385, −6.377385262845885, −6.141819539785096, −5.660492390430629, −4.883578815312258, −4.375676691423881, −3.970604614844386, −3.546479851458495, −2.919075063110200, −2.364920408039468, −1.782146706814168, −0.9200173783783240, −0.3646520482789482,
0.3646520482789482, 0.9200173783783240, 1.782146706814168, 2.364920408039468, 2.919075063110200, 3.546479851458495, 3.970604614844386, 4.375676691423881, 4.883578815312258, 5.660492390430629, 6.141819539785096, 6.377385262845885, 6.841965700006385, 7.724622126299574, 7.996045490445856, 8.391702617364969, 8.747734111669649, 9.224893428565536, 9.710526821649313, 10.55513614447631, 10.82476118035061, 11.17220742043455, 11.44018880510507, 12.09830317416286, 12.43853008318337
