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
−15.28236477429743964757375171696, −14.02747758985131441322366072081, −12.80479661881926758434461198076, −11.65476666862771273517783541090, −10.08229835157541151093983457869, −9.549608680332154597170236147059, −8.063876026994590903985309774918, −6.72499890221612177908178142859, −4.70711555479403755874564285250, −3.56195934686253984012445182266,
0.34604397852190797751174751921, 2.42974270888537115205070170647, 5.31964949110959194727223828339, 6.29944203610274301509320317221, 8.018992664988703403517023652807, 9.148625703392210982180893149484, 10.40445310786238657933304399976, 11.70458655955200457844591288999, 12.66501928899394734763551439444, 13.84195414159230340533329530259