| L(s) = 1 | − 4·2-s + (14.0 + 6.82i)3-s + 16·4-s − 34.7i·5-s + (−56.0 − 27.2i)6-s − 97.2·7-s − 64·8-s + (149. + 191. i)9-s + 139. i·10-s + 627.·11-s + (224. + 109. i)12-s + 675. i·13-s + 389.·14-s + (237. − 487. i)15-s + 256·16-s − 560. i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (0.899 + 0.437i)3-s + 0.5·4-s − 0.622i·5-s + (−0.635 − 0.309i)6-s − 0.750·7-s − 0.353·8-s + (0.616 + 0.787i)9-s + 0.439i·10-s + 1.56·11-s + (0.449 + 0.218i)12-s + 1.10i·13-s + 0.530·14-s + (0.272 − 0.559i)15-s + 0.250·16-s − 0.470i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.829424250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.829424250\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + (-14.0 - 6.82i)T \) |
| 59 | \( 1 + (-4.95e3 - 2.62e4i)T \) |
| good | 5 | \( 1 + 34.7iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 97.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 627.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 675. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 560. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.62e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 334.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.74e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 8.35e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.56e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 7.86e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.29e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.12e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.00e3iT - 4.18e8T^{2} \) |
| 61 | \( 1 - 2.69e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.55e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.55e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 4.51e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 9.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.14e4iT - 8.58e9T^{2} \) |
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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
−10.52216579292087353509145032117, −9.611619633576276316145536353155, −8.934048980298245299486809060124, −8.539650438089443090553529947950, −7.06750278775810102497233196564, −6.38491179597534196311506140038, −4.59778988133809764591036184409, −3.73414705562015635726857029216, −2.34470758249318632775104506988, −1.16071495262547113281396890968,
0.56873160917921006956439053481, 1.91779252407448124108830645624, 3.06551703695747490691014278384, 3.95927518149341193456613155487, 6.18516323818572518525894121683, 6.70852303580984861204544372668, 7.68943693859149366724420101324, 8.710863711364787739901500628941, 9.332050977047014098086153828480, 10.30981950323829287597318823078
