| L(s) = 1 | + (−1.47 + 1.47i)3-s + (55.2 − 8.52i)5-s + (−156. − 156. i)7-s + 238. i·9-s − 35.4i·11-s + (−247. − 247. i)13-s + (−69.0 + 94.2i)15-s + (−1.19e3 + 1.19e3i)17-s + 1.01e3·19-s + 461.·21-s + (−2.03e3 + 2.03e3i)23-s + (2.97e3 − 942. i)25-s + (−711. − 711. i)27-s + 2.20e3i·29-s − 6.17e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.0947 + 0.0947i)3-s + (0.988 − 0.152i)5-s + (−1.20 − 1.20i)7-s + 0.982i·9-s − 0.0884i·11-s + (−0.405 − 0.405i)13-s + (−0.0792 + 0.108i)15-s + (−1.00 + 1.00i)17-s + 0.642·19-s + 0.228·21-s + (−0.801 + 0.801i)23-s + (0.953 − 0.301i)25-s + (−0.187 − 0.187i)27-s + 0.487i·29-s − 1.15i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.389i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2374671523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2374671523\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-55.2 + 8.52i)T \) |
| good | 3 | \( 1 + (1.47 - 1.47i)T - 243iT^{2} \) |
| 7 | \( 1 + (156. + 156. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 + 35.4iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (247. + 247. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (1.19e3 - 1.19e3i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 1.01e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (2.03e3 - 2.03e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 2.20e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 6.17e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (9.46e3 - 9.46e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 9.00e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (1.59e4 - 1.59e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (7.19e3 + 7.19e3i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.29e4 + 1.29e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 4.05e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.92e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (1.82e4 + 1.82e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.66e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-3.64e4 - 3.64e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 - 5.08e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + (2.03e4 - 2.03e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 5.93e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.95e4 + 6.95e4i)T - 8.58e9iT^{2} \) |
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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
−12.78637249678643814345032526454, −11.19916501448478482277330168501, −10.11491033528489129233198805102, −9.820188373928452743027025486318, −8.282916211813591987146201588446, −7.02722749640854666740779735977, −6.04730905960715215816522095424, −4.77911057998266451363415720443, −3.31433785913348425022763568765, −1.71581270691586988494395602442,
0.07221184531619248884390157712, 2.12221181455902379179168657359, 3.23333851799923825318488979954, 5.15115103999423719844328401273, 6.28701718008943934244511992950, 6.86318748082079971225708406956, 8.930170222564079729725589477610, 9.350893098948352798790422206126, 10.29747843756993399224674348910, 11.87581199063583750684533405601
