| L(s) = 1 | + (−0.998 − 0.576i)2-s + (−1.29 − 5.03i)3-s + (−3.33 − 5.77i)4-s + 0.274·5-s + (−1.60 + 5.77i)6-s + (−9.15 + 16.1i)7-s + 16.9i·8-s + (−23.6 + 13.0i)9-s + (−0.274 − 0.158i)10-s − 26.2i·11-s + (−24.7 + 24.2i)12-s + (−39.6 − 22.8i)13-s + (18.4 − 10.7i)14-s + (−0.356 − 1.38i)15-s + (−16.9 + 29.3i)16-s + (30.2 − 52.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.352 − 0.203i)2-s + (−0.250 − 0.968i)3-s + (−0.416 − 0.722i)4-s + 0.0245·5-s + (−0.109 + 0.392i)6-s + (−0.494 + 0.869i)7-s + 0.747i·8-s + (−0.874 + 0.484i)9-s + (−0.00867 − 0.00500i)10-s − 0.719i·11-s + (−0.595 + 0.584i)12-s + (−0.845 − 0.487i)13-s + (0.351 − 0.206i)14-s + (−0.00614 − 0.0237i)15-s + (−0.264 + 0.458i)16-s + (0.431 − 0.747i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0522529 + 0.444825i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0522529 + 0.444825i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.29 + 5.03i)T \) |
| 7 | \( 1 + (9.15 - 16.1i)T \) |
| good | 2 | \( 1 + (0.998 + 0.576i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 0.274T + 125T^{2} \) |
| 11 | \( 1 + 26.2iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (39.6 + 22.8i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-30.2 + 52.3i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (38.8 - 22.4i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 127. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-58.2 + 33.6i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-85.6 + 49.4i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (123. + 214. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-134. + 232. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (72.4 + 125. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (250. - 434. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (333. + 192. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-274. - 475. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-189. - 109. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-378. - 656. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 1.10e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (125. + 72.6i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-445. + 772. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-745. - 1.29e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (145. + 251. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-144. + 83.6i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.84195414159230340533329530259, −12.66501928899394734763551439444, −11.70458655955200457844591288999, −10.40445310786238657933304399976, −9.148625703392210982180893149484, −8.018992664988703403517023652807, −6.29944203610274301509320317221, −5.31964949110959194727223828339, −2.42974270888537115205070170647, −0.34604397852190797751174751921,
3.56195934686253984012445182266, 4.70711555479403755874564285250, 6.72499890221612177908178142859, 8.063876026994590903985309774918, 9.549608680332154597170236147059, 10.08229835157541151093983457869, 11.65476666862771273517783541090, 12.80479661881926758434461198076, 14.02747758985131441322366072081, 15.28236477429743964757375171696
