L(s) = 1 | + (−1.87 − 0.684i)2-s + (6.42 − 2.33i)3-s + (3.06 + 2.57i)4-s + (−3.55 + 20.1i)5-s − 13.6·6-s + (−4.31 + 24.4i)7-s + (−4.00 − 6.92i)8-s + (15.1 − 12.6i)9-s + (20.4 − 35.4i)10-s + (−11.5 − 19.9i)11-s + (25.6 + 9.35i)12-s + (11.6 + 9.73i)13-s + (24.8 − 43.0i)14-s + (24.3 + 137. i)15-s + (2.77 + 15.7i)16-s + (−22.3 + 18.7i)17-s + ⋯ |
L(s) = 1 | + (−0.664 − 0.241i)2-s + (1.23 − 0.449i)3-s + (0.383 + 0.321i)4-s + (−0.318 + 1.80i)5-s − 0.930·6-s + (−0.233 + 1.32i)7-s + (−0.176 − 0.306i)8-s + (0.559 − 0.469i)9-s + (0.648 − 1.12i)10-s + (−0.315 − 0.546i)11-s + (0.618 + 0.224i)12-s + (0.247 + 0.207i)13-s + (0.474 − 0.822i)14-s + (0.418 + 2.37i)15-s + (0.0434 + 0.246i)16-s + (−0.318 + 0.267i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.28228 + 0.625308i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28228 + 0.625308i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.87 + 0.684i)T \) |
| 37 | \( 1 + (-222. + 33.9i)T \) |
good | 3 | \( 1 + (-6.42 + 2.33i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (3.55 - 20.1i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (4.31 - 24.4i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (11.5 + 19.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-11.6 - 9.73i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (22.3 - 18.7i)T + (853. - 4.83e3i)T^{2} \) |
| 19 | \( 1 + (-145. + 52.8i)T + (5.25e3 - 4.40e3i)T^{2} \) |
| 23 | \( 1 + (-91.8 + 159. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (24.9 + 43.2i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 171.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (50.7 + 42.5i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + 22.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + (242. - 419. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (0.712 + 4.04i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-45.6 - 258. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-137. - 115. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-65.9 + 373. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (896. - 326. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 - 733.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-93.2 + 529. i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-574. + 482. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (124. + 707. i)T + (-6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (189. - 328. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.38381022835992298128341200487, −13.30202682663045407584949839728, −11.80164277343442075421794089670, −10.89085029482906976240972225026, −9.519391419409266189235483122288, −8.483195384174788141688304799681, −7.46276801987943370007777919097, −6.31048993883379948895049611116, −3.05815456077191075036462489228, −2.58662138128592372200852831257,
1.09036366177258325475959864173, 3.65327608080925891778463192900, 5.05449550937566126913814645713, 7.46847098092942882726440439640, 8.202189055564018610509109217148, 9.357627374311410879991123154983, 9.907201496960131574379346907065, 11.67361242573472457646473153664, 13.17475555657980771359390570781, 13.80963681611100603514347685812