| L(s) = 1 | + (−0.667 + 1.15i)2-s + (5.07 + 1.13i)3-s + (3.10 + 5.38i)4-s + 9.00·5-s + (−4.69 + 5.11i)6-s + (−14.2 − 11.8i)7-s − 18.9·8-s + (24.4 + 11.4i)9-s + (−6.01 + 10.4i)10-s + 29.3·11-s + (9.67 + 30.8i)12-s + (−21.1 + 36.6i)13-s + (23.2 − 8.53i)14-s + (45.6 + 10.1i)15-s + (−12.1 + 21.1i)16-s + (−2.56 + 4.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.236 + 0.408i)2-s + (0.976 + 0.217i)3-s + (0.388 + 0.672i)4-s + 0.805·5-s + (−0.319 + 0.347i)6-s + (−0.767 − 0.640i)7-s − 0.839·8-s + (0.905 + 0.424i)9-s + (−0.190 + 0.329i)10-s + 0.804·11-s + (0.232 + 0.741i)12-s + (−0.451 + 0.782i)13-s + (0.443 − 0.162i)14-s + (0.786 + 0.175i)15-s + (−0.190 + 0.329i)16-s + (−0.0366 + 0.0634i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.877i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.478 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.63226 + 0.969040i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63226 + 0.969040i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.07 - 1.13i)T \) |
| 7 | \( 1 + (14.2 + 11.8i)T \) |
| good | 2 | \( 1 + (0.667 - 1.15i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 - 9.00T + 125T^{2} \) |
| 11 | \( 1 - 29.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + (21.1 - 36.6i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (2.56 - 4.45i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (71.2 + 123. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 - 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (109. + 189. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (73.9 + 128. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (21.2 + 36.8i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-83.7 + 145. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-121. - 210. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-38.2 + 66.2i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (181. - 314. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-60.7 - 105. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-321. + 556. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-81.4 - 141. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 833.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (62.4 - 108. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (421. - 729. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-566. - 982. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (248. + 429. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (128. + 223. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.73797409350146935569539003561, −13.53206428463030612097546667944, −12.82654126646545926040951263544, −11.15519843719933661949077863264, −9.541672247383449265251166458397, −9.004787885880322767835422200126, −7.37471354995253659640959387616, −6.52551834922756235785170814536, −4.07206979727604364915410993316, −2.49223275623022029609585675528,
1.71509496486171512156334235166, 3.12488559770882272315155764445, 5.71998777445562615803054034389, 6.90429159397109226772593981996, 8.815371681663048570408147891627, 9.595207877982747602679576942038, 10.50888986762320150110850570057, 12.23582526249607899279555994021, 13.08368104231258349420382234320, 14.53145209140676205919321556246
