| L(s) = 1 | + (−0.732 − 2.73i)2-s + (−13.9 + 8.07i)3-s + (−6.92 + 4i)4-s + (−9.30 + 34.7i)5-s + (32.3 + 32.3i)6-s + (−30.9 − 53.5i)7-s + (16 + 15.9i)8-s + (90.0 − 155. i)9-s + 101.·10-s + 185. i·11-s + (64.6 − 111. i)12-s + (72.0 − 268. i)13-s + (−123. + 123. i)14-s + (−150. − 561. i)15-s + (31.9 − 55.4i)16-s + (−102. + 27.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 − 0.683i)2-s + (−1.55 + 0.897i)3-s + (−0.433 + 0.250i)4-s + (−0.372 + 1.38i)5-s + (0.897 + 0.897i)6-s + (−0.631 − 1.09i)7-s + (0.250 + 0.249i)8-s + (1.11 − 1.92i)9-s + 1.01·10-s + 1.53i·11-s + (0.448 − 0.777i)12-s + (0.426 − 1.59i)13-s + (−0.631 + 0.631i)14-s + (−0.668 − 2.49i)15-s + (0.124 − 0.216i)16-s + (−0.355 + 0.0951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0656 + 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0656 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.259396 - 0.242902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.259396 - 0.242902i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.732 + 2.73i)T \) |
| 37 | \( 1 + (-1.35e3 - 223. i)T \) |
| good | 3 | \( 1 + (13.9 - 8.07i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (9.30 - 34.7i)T + (-541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (30.9 + 53.5i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 185. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-72.0 + 268. i)T + (-2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (102. - 27.5i)T + (7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-2.22 + 8.32i)T + (-1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-46.1 - 46.1i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (-734. + 734. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (-78.9 + 78.9i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (266. - 153. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (2.03e3 + 2.03e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 821.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-1.20e3 + 2.08e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (5.75e3 - 1.54e3i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (813. + 218. i)T + (1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-1.43e3 + 830. i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (3.04e3 + 5.27e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 - 49.7iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (412. - 1.53e3i)T + (-3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-6.30e3 + 1.09e4i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (3.11e3 + 1.16e4i)T + (-5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (651. + 651. i)T + 8.85e7iT^{2} \) |
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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
−13.23750746314921047277053287831, −12.09000253355472781465471873698, −11.04838671245789296044510301084, −10.25462665280796177579857496057, −10.01107672887759822978580572355, −7.40656903462346324473186467104, −6.32346522650859215887958888469, −4.57588222032233064032788894863, −3.39564865715704793652858520161, −0.29499405048036909162538670049,
1.10525416417495829286258848346, 4.75615751761935234150069825074, 5.86929724237167141998916207389, 6.60080724349675460049605241480, 8.312769050205278328562602836276, 9.173150892890067378346915437983, 11.18268691887877759959366664730, 11.99879021846899440143605651761, 12.83493465150907012895929472119, 13.75755546070399758676701235391
