| L(s) = 1 | + 2·5-s + 4·11-s + 6·13-s + 2·17-s + 4·19-s + 8·23-s − 25-s − 2·29-s + 10·37-s − 6·41-s − 4·43-s + 6·53-s + 8·55-s + 4·59-s + 6·61-s + 12·65-s + 4·67-s + 8·71-s − 10·73-s − 4·83-s + 4·85-s − 6·89-s + 8·95-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.371·29-s + 1.64·37-s − 0.937·41-s − 0.609·43-s + 0.824·53-s + 1.07·55-s + 0.520·59-s + 0.768·61-s + 1.48·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s − 0.439·83-s + 0.433·85-s − 0.635·89-s + 0.820·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.380018149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.380018149\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.21388562812125, −14.51257185227895, −14.22911067343575, −13.45338717934197, −13.29381831402501, −12.72735657666329, −11.82949418965155, −11.42522122936979, −11.08243937382584, −10.19175655828558, −9.830267974455839, −9.122785198265599, −8.869988892338325, −8.168770677969363, −7.414240107640746, −6.721973763207813, −6.316133508638844, −5.661210930275668, −5.234074038591058, −4.311560087919987, −3.618245860922687, −3.149291213774677, −2.177541959622044, −1.274080513607606, −0.9882400813330538,
0.9882400813330538, 1.274080513607606, 2.177541959622044, 3.149291213774677, 3.618245860922687, 4.311560087919987, 5.234074038591058, 5.661210930275668, 6.316133508638844, 6.721973763207813, 7.414240107640746, 8.168770677969363, 8.869988892338325, 9.122785198265599, 9.830267974455839, 10.19175655828558, 11.08243937382584, 11.42522122936979, 11.82949418965155, 12.72735657666329, 13.29381831402501, 13.45338717934197, 14.22911067343575, 14.51257185227895, 15.21388562812125
