| L(s) = 1 | + (2.97 − 1.51i)2-s + (15.7 − 2.49i)3-s + (−2.84 + 3.91i)4-s + (−22.1 + 11.5i)5-s + (43.1 − 31.3i)6-s + (−44.2 − 44.2i)7-s + (−10.8 + 68.7i)8-s + (165. − 53.8i)9-s + (−48.5 + 67.9i)10-s + (18.0 − 55.5i)11-s + (−35.0 + 68.8i)12-s + (−96.0 − 48.9i)13-s + (−198. − 64.6i)14-s + (−321. + 237. i)15-s + (47.9 + 147. i)16-s + (39.1 + 6.19i)17-s + ⋯ |
| L(s) = 1 | + (0.744 − 0.379i)2-s + (1.75 − 0.277i)3-s + (−0.177 + 0.244i)4-s + (−0.887 + 0.461i)5-s + (1.19 − 0.871i)6-s + (−0.903 − 0.903i)7-s + (−0.170 + 1.07i)8-s + (2.04 − 0.664i)9-s + (−0.485 + 0.679i)10-s + (0.149 − 0.459i)11-s + (−0.243 + 0.477i)12-s + (−0.568 − 0.289i)13-s + (−1.01 − 0.329i)14-s + (−1.42 + 1.05i)15-s + (0.187 + 0.576i)16-s + (0.135 + 0.0214i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.29847 - 0.418281i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29847 - 0.418281i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (22.1 - 11.5i)T \) |
| good | 2 | \( 1 + (-2.97 + 1.51i)T + (9.40 - 12.9i)T^{2} \) |
| 3 | \( 1 + (-15.7 + 2.49i)T + (77.0 - 25.0i)T^{2} \) |
| 7 | \( 1 + (44.2 + 44.2i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + (-18.0 + 55.5i)T + (-1.18e4 - 8.60e3i)T^{2} \) |
| 13 | \( 1 + (96.0 + 48.9i)T + (1.67e4 + 2.31e4i)T^{2} \) |
| 17 | \( 1 + (-39.1 - 6.19i)T + (7.94e4 + 2.58e4i)T^{2} \) |
| 19 | \( 1 + (-346. - 477. i)T + (-4.02e4 + 1.23e5i)T^{2} \) |
| 23 | \( 1 + (-66.0 - 129. i)T + (-1.64e5 + 2.26e5i)T^{2} \) |
| 29 | \( 1 + (-361. + 497. i)T + (-2.18e5 - 6.72e5i)T^{2} \) |
| 31 | \( 1 + (227. - 165. i)T + (2.85e5 - 8.78e5i)T^{2} \) |
| 37 | \( 1 + (60.5 - 118. i)T + (-1.10e6 - 1.51e6i)T^{2} \) |
| 41 | \( 1 + (152. + 468. i)T + (-2.28e6 + 1.66e6i)T^{2} \) |
| 43 | \( 1 + (1.71e3 - 1.71e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (559. + 3.53e3i)T + (-4.64e6 + 1.50e6i)T^{2} \) |
| 53 | \( 1 + (-2.21e3 + 350. i)T + (7.50e6 - 2.43e6i)T^{2} \) |
| 59 | \( 1 + (-461. + 150. i)T + (9.80e6 - 7.12e6i)T^{2} \) |
| 61 | \( 1 + (1.34e3 - 4.14e3i)T + (-1.12e7 - 8.13e6i)T^{2} \) |
| 67 | \( 1 + (3.64e3 + 577. i)T + (1.91e7 + 6.22e6i)T^{2} \) |
| 71 | \( 1 + (-2.69e3 - 1.95e3i)T + (7.85e6 + 2.41e7i)T^{2} \) |
| 73 | \( 1 + (1.28e3 + 2.52e3i)T + (-1.66e7 + 2.29e7i)T^{2} \) |
| 79 | \( 1 + (3.55e3 - 4.89e3i)T + (-1.20e7 - 3.70e7i)T^{2} \) |
| 83 | \( 1 + (485. - 3.06e3i)T + (-4.51e7 - 1.46e7i)T^{2} \) |
| 89 | \( 1 + (2.69e3 + 875. i)T + (5.07e7 + 3.68e7i)T^{2} \) |
| 97 | \( 1 + (-811. - 5.12e3i)T + (-8.41e7 + 2.73e7i)T^{2} \) |
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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
−16.44192790016112799098028495210, −14.97321495647283972145253405534, −14.04877706158244343823571952403, −13.24392694242174315756888578210, −12.04129000738600341529271683498, −9.999994293848972016369739163740, −8.328306560906517489300916699736, −7.31517205886866826066970359816, −3.84765153079632187783726251961, −3.10154296291252124708412097119,
3.17589126084729234071628269260, 4.70783209902508982571668493070, 7.17451539789907503347921511415, 8.897722129872571556469099070969, 9.640377083181055328411619435981, 12.36262798874449941602440137689, 13.34468647402028148928398549497, 14.57028538698065369067962791821, 15.41034874654977496873954625754, 16.01151911905165316329976656447
