| 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
−15.45851727808528981303886244236, −14.82580875195637485583118076692, −13.45927890177749757674156314637, −11.63743632200873638437614924670, −11.30041518723569478699093951634, −9.322887966082246590255056504611, −8.365904127573809087843182485119, −6.58140139552612933754561370008, −5.18109777596533364593029826242, −3.89813038790197817982660086780,
1.28167547908487353258785614270, 3.93470924510749972638777092112, 5.67891266928638997498994804176, 7.39867821920338982363694568641, 8.545137533830433208259625900419, 10.60337495204715213256352011606, 11.40978197367245592930860897047, 12.27844126474603302129337175249, 13.56532565300236507765883527232, 14.27194393316916272725625014142
