| L(s) = 1 | − 1.61i·2-s − 1.23i·3-s − 0.618·4-s − 2.00·6-s − 2.23i·8-s + 1.47·9-s + 4.23·11-s + 0.763i·12-s + 3.23i·13-s − 4.85·16-s − 6.47i·17-s − 2.38i·18-s + 4.47·19-s − 6.85i·22-s + 1.76i·23-s − 2.76·24-s + ⋯ |
| L(s) = 1 | − 1.14i·2-s − 0.713i·3-s − 0.309·4-s − 0.816·6-s − 0.790i·8-s + 0.490·9-s + 1.27·11-s + 0.220i·12-s + 0.897i·13-s − 1.21·16-s − 1.56i·17-s − 0.561i·18-s + 1.02·19-s − 1.46i·22-s + 0.367i·23-s − 0.564·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.026365973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.026365973\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.61iT - 2T^{2} \) |
| 3 | \( 1 + 1.23iT - 3T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 + 6.47iT - 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 1.76iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 9.70T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 + 9.23T + 41T^{2} \) |
| 43 | \( 1 - 6.23iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 0.472iT - 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 + 0.236iT - 67T^{2} \) |
| 71 | \( 1 + 4.70T + 71T^{2} \) |
| 73 | \( 1 - 13.2iT - 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 5.70iT - 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 0.763iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.606606134920381213734462127963, −8.890023287700151269741865039725, −7.50858109766352081726063315505, −6.96500513448782911684694554939, −6.28322838270318941266615162931, −4.80024566876612820989611811789, −3.90958486544072445413140700674, −2.89428874118669310415636588789, −1.76600401765979428066316529703, −0.981588788190884553490601443660,
1.56446259368948466073359165452, 3.26315895062392405127863435071, 4.21520229611882138490212393104, 5.11337528885542241145017963819, 6.01913710442662557314706496631, 6.68024070795720090866326855678, 7.56790463211620013017855356132, 8.355476162824319135873362421038, 9.082794095015426583749281858105, 10.02625082937657844739980190277
