L(s) = 1 | + (−5.49 + 9.52i)5-s + (6.85 − 17.2i)7-s + (−13.8 + 7.98i)11-s + (−77.4 − 44.7i)13-s + 106.·17-s − 8.31i·19-s + (123. + 71.5i)23-s + (2.06 + 3.58i)25-s + (−129. + 74.8i)29-s + (−37.1 − 21.4i)31-s + (126. + 159. i)35-s + 390.·37-s + (−172. + 298. i)41-s + (28.5 + 49.4i)43-s + (−8.13 − 14.0i)47-s + ⋯ |
L(s) = 1 | + (−0.491 + 0.851i)5-s + (0.370 − 0.928i)7-s + (−0.378 + 0.218i)11-s + (−1.65 − 0.953i)13-s + 1.52·17-s − 0.100i·19-s + (1.12 + 0.648i)23-s + (0.0165 + 0.0286i)25-s + (−0.830 + 0.479i)29-s + (−0.215 − 0.124i)31-s + (0.609 + 0.772i)35-s + 1.73·37-s + (−0.657 + 1.13i)41-s + (0.101 + 0.175i)43-s + (−0.0252 − 0.0437i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 - 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.297192640\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.297192640\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-6.85 + 17.2i)T \) |
good | 5 | \( 1 + (5.49 - 9.52i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (13.8 - 7.98i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (77.4 + 44.7i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 8.31iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-123. - 71.5i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (129. - 74.8i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (37.1 + 21.4i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 390.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (172. - 298. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-28.5 - 49.4i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (8.13 + 14.0i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 445. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (193. - 334. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (420. - 242. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (251. - 436. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 751. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 507. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-381. - 660. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-607. - 1.05e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 425.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-494. + 285. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.11067888109808115973704986250, −9.632900203614562816590999305482, −8.054247279052588576325349095406, −7.46167362060802749587962371050, −7.06840218647774164711129611130, −5.56085839532045419782093625413, −4.76466812681626726712253737712, −3.49651646874149505063786851531, −2.73084475150361620076634219117, −1.02723832819527432230039821973,
0.41538388264360298633533894919, 1.91609411681371066622705774915, 3.06006330145209946708907416766, 4.55059134172288191493394643649, 5.07210281997027963480609630381, 6.06640161142353361321588463099, 7.43483515882046322647574251853, 7.967823505080250185455303717856, 9.031260357228243162692775867821, 9.478102114534421172559302032025