| L(s) = 1 | + (−2.73 − 0.732i)2-s + (6.38 + 3.68i)3-s + (6.92 + 4i)4-s + (−43.6 + 11.6i)5-s + (−14.7 − 14.7i)6-s + (39.7 − 68.7i)7-s + (−15.9 − 16i)8-s + (−13.3 − 23.0i)9-s + 127.·10-s + 23.6i·11-s + (29.4 + 51.0i)12-s + (240. − 64.4i)13-s + (−158. + 158. i)14-s + (−321. − 86.1i)15-s + (31.9 + 55.4i)16-s + (122. − 457. i)17-s + ⋯ |
| L(s) = 1 | + (−0.683 − 0.183i)2-s + (0.709 + 0.409i)3-s + (0.433 + 0.250i)4-s + (−1.74 + 0.467i)5-s + (−0.409 − 0.409i)6-s + (0.810 − 1.40i)7-s + (−0.249 − 0.250i)8-s + (−0.164 − 0.284i)9-s + 1.27·10-s + 0.195i·11-s + (0.204 + 0.354i)12-s + (1.42 − 0.381i)13-s + (−0.810 + 0.810i)14-s + (−1.42 − 0.382i)15-s + (0.124 + 0.216i)16-s + (0.424 − 1.58i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.723001 - 0.648611i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.723001 - 0.648611i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.73 + 0.732i)T \) |
| 37 | \( 1 + (1.34e3 - 235. i)T \) |
| good | 3 | \( 1 + (-6.38 - 3.68i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (43.6 - 11.6i)T + (541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (-39.7 + 68.7i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 - 23.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-240. + 64.4i)T + (2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-122. + 457. i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (126. - 33.9i)T + (1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (496. + 496. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (503. - 503. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (358. - 358. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (-646. - 373. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (553. + 553. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 - 2.64e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (104. + 180. i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-57.1 + 213. i)T + (-1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-1.12e3 - 4.21e3i)T + (-1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-1.83e3 - 1.05e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-1.61e3 + 2.79e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + 9.19e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (1.06e4 - 2.84e3i)T + (3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-2.78e3 - 4.82e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-1.02e3 - 274. i)T + (5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-6.80e3 - 6.80e3i)T + 8.85e7iT^{2} \) |
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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
−13.89459519581753765566529624464, −12.07697465229090398976229247597, −11.13128843376744807479680327247, −10.39405709097433782645975877748, −8.729129534717936939343891292407, −7.908009924316951160700775381123, −7.04098731661657329395342858130, −4.18802198048966448444242356071, −3.34049633149160564438715792500, −0.60681424499263306771683518388,
1.75400552665590181471299092853, 3.76775091816120699360132022613, 5.73657128254103296906605303090, 7.69709570826854877523317928359, 8.389198101759155483629370888766, 8.796505658124125966290652362858, 11.03414514602682148828756942080, 11.70948176154801120068705660238, 12.80457151304002591020516817113, 14.38876373766340882746498036472
