| L(s) = 1 | + (2.73 + 0.732i)2-s + (13.1 − 3.51i)3-s + (6.92 + 4i)4-s + (24.5 + 4.80i)5-s + 38.4·6-s + (−43.1 − 23.2i)7-s + (15.9 + 16i)8-s + (89.7 − 51.8i)9-s + (63.5 + 31.0i)10-s + (−88.7 + 153. i)11-s + (105. + 28.1i)12-s + (−199. − 199. i)13-s + (−100. − 95.0i)14-s + (338. − 23.2i)15-s + (31.9 + 55.4i)16-s + (14.1 + 52.7i)17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.183i)2-s + (1.45 − 0.390i)3-s + (0.433 + 0.250i)4-s + (0.981 + 0.192i)5-s + 1.06·6-s + (−0.880 − 0.474i)7-s + (0.249 + 0.250i)8-s + (1.10 − 0.639i)9-s + (0.635 + 0.310i)10-s + (−0.733 + 1.27i)11-s + (0.729 + 0.195i)12-s + (−1.18 − 1.18i)13-s + (−0.514 − 0.485i)14-s + (1.50 − 0.103i)15-s + (0.124 + 0.216i)16-s + (0.0489 + 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00446i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.00446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.67883 + 0.00821722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.67883 + 0.00821722i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.73 - 0.732i)T \) |
| 5 | \( 1 + (-24.5 - 4.80i)T \) |
| 7 | \( 1 + (43.1 + 23.2i)T \) |
| good | 3 | \( 1 + (-13.1 + 3.51i)T + (70.1 - 40.5i)T^{2} \) |
| 11 | \( 1 + (88.7 - 153. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (199. + 199. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-14.1 - 52.7i)T + (-7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-353. + 203. i)T + (6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (181. - 677. i)T + (-2.42e5 - 1.39e5i)T^{2} \) |
| 29 | \( 1 + 153. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + (-477. + 827. i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (457. + 122. i)T + (1.62e6 + 9.37e5i)T^{2} \) |
| 41 | \( 1 + 2.06e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (1.42e3 + 1.42e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (-160. - 43.0i)T + (4.22e6 + 2.43e6i)T^{2} \) |
| 53 | \( 1 + (1.38e3 - 369. i)T + (6.83e6 - 3.94e6i)T^{2} \) |
| 59 | \( 1 + (-5.02e3 - 2.90e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (603. + 1.04e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (474. + 1.77e3i)T + (-1.74e7 + 1.00e7i)T^{2} \) |
| 71 | \( 1 - 4.56e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (637. - 170. i)T + (2.45e7 - 1.41e7i)T^{2} \) |
| 79 | \( 1 + (-9.81e3 + 5.66e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-5.70e3 - 5.70e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + (2.68e3 - 1.55e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-4.38e3 + 4.38e3i)T - 8.85e7iT^{2} \) |
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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
−13.68915555141438772986775615464, −13.30745338825499871195125333830, −12.37936175053540696626637960092, −10.13247989558474860521554703524, −9.574129084225500029054029076697, −7.75865668535764352121504262001, −7.01835482623321997875114837185, −5.26030367570711345701853989549, −3.25438671749010487359236563216, −2.23861860768965633276924690926,
2.32856167655369549355404834488, 3.27631499868866198529878944758, 5.10230551017340684328400483675, 6.59165449642778502347995735264, 8.399348124475292266990817922630, 9.467048301913741189236742729740, 10.22142960072620574914619858232, 12.07876263671513121941106128362, 13.25354165423716953963233761258, 13.99249696326805998576180326452
