| L(s) = 1 | + (0.483 − 1.32i)2-s + (−1.47 − 2.61i)3-s + (−1.53 − 1.28i)4-s + (−2.99 − 0.528i)5-s + (−4.18 + 0.693i)6-s + (6.30 − 5.28i)7-s + (−2.44 + 1.41i)8-s + (−4.66 + 7.69i)9-s + (−2.15 + 3.72i)10-s + (6.45 − 1.13i)11-s + (−1.10 + 5.89i)12-s + (17.4 − 6.33i)13-s + (−3.98 − 10.9i)14-s + (3.03 + 8.60i)15-s + (0.694 + 3.93i)16-s + (−11.4 − 6.59i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.664i)2-s + (−0.490 − 0.871i)3-s + (−0.383 − 0.321i)4-s + (−0.599 − 0.105i)5-s + (−0.697 + 0.115i)6-s + (0.900 − 0.755i)7-s + (−0.306 + 0.176i)8-s + (−0.517 + 0.855i)9-s + (−0.215 + 0.372i)10-s + (0.586 − 0.103i)11-s + (−0.0919 + 0.491i)12-s + (1.33 − 0.487i)13-s + (−0.284 − 0.781i)14-s + (0.202 + 0.573i)15-s + (0.0434 + 0.246i)16-s + (−0.672 − 0.388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.545351 - 0.915712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.545351 - 0.915712i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.483 + 1.32i)T \) |
| 3 | \( 1 + (1.47 + 2.61i)T \) |
| good | 5 | \( 1 + (2.99 + 0.528i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-6.30 + 5.28i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-6.45 + 1.13i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-17.4 + 6.33i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (11.4 + 6.59i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-17.4 - 30.1i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-2.12 + 2.53i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (9.72 - 26.7i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (10.6 + 8.92i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (3.33 - 5.77i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (19.4 + 53.5i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (5.68 + 32.2i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-34.5 - 41.1i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 98.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-101. - 17.8i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (5.50 - 4.61i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (28.2 - 10.2i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-8.46 - 4.88i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-64.7 - 112. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (96.4 + 35.1i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-37.0 + 101. i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (20.1 - 11.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (8.89 + 50.4i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−14.27194393316916272725625014142, −13.56532565300236507765883527232, −12.27844126474603302129337175249, −11.40978197367245592930860897047, −10.60337495204715213256352011606, −8.545137533830433208259625900419, −7.39867821920338982363694568641, −5.67891266928638997498994804176, −3.93470924510749972638777092112, −1.28167547908487353258785614270,
3.89813038790197817982660086780, 5.18109777596533364593029826242, 6.58140139552612933754561370008, 8.365904127573809087843182485119, 9.322887966082246590255056504611, 11.30041518723569478699093951634, 11.63743632200873638437614924670, 13.45927890177749757674156314637, 14.82580875195637485583118076692, 15.45851727808528981303886244236
