L(s) = 1 | + (−3.91 + 2.26i)2-s + (6.22 − 10.7i)4-s + (0.632 + 1.09i)5-s + 20.1i·8-s + (−4.95 − 2.86i)10-s + (36.0 + 20.7i)11-s − 85.7i·13-s + (4.27 + 7.39i)16-s + (−38.8 + 67.3i)17-s + (−42.1 + 24.3i)19-s + 15.7·20-s − 188.·22-s + (78.7 − 45.4i)23-s + (61.6 − 106. i)25-s + (193. + 335. i)26-s + ⋯ |
L(s) = 1 | + (−1.38 + 0.799i)2-s + (0.778 − 1.34i)4-s + (0.0566 + 0.0980i)5-s + 0.890i·8-s + (−0.156 − 0.0905i)10-s + (0.987 + 0.570i)11-s − 1.82i·13-s + (0.0667 + 0.115i)16-s + (−0.554 + 0.961i)17-s + (−0.509 + 0.293i)19-s + 0.176·20-s − 1.82·22-s + (0.714 − 0.412i)23-s + (0.493 − 0.854i)25-s + (1.46 + 2.53i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6614819882\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6614819882\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (3.91 - 2.26i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-0.632 - 1.09i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-36.0 - 20.7i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 85.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (38.8 - 67.3i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (42.1 - 24.3i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-78.7 + 45.4i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 151. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (76.3 + 44.0i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (45.2 + 78.4i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 383.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 227.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (69.5 + 120. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (289. + 167. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-440. + 762. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (11.3 - 6.57i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-221. + 383. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 341. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-798. - 460. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-206. - 357. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 954.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-14.8 - 25.7i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.19e3iT - 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.39211006765497974064010210166, −9.605167564433087778903462252296, −8.555267802511282310935954318853, −8.134542575587845168300892554899, −6.91426930939092796795416796274, −6.38180735755859735660111534015, −5.14223358155300151759586723811, −3.55854158807932015343777949481, −1.76638808693315432262294070571, −0.39926928109252492405490738615,
1.10949179434415875684113430141, 2.16072464017595473136054007757, 3.52032478365351937179677829588, 4.85660397513698427082678450176, 6.54077010408202654980242886852, 7.23309981282868927389045938349, 8.591800894186639078630778155119, 9.099851938657947808053443096713, 9.651286295969634552529118890065, 10.85364998751055096158229644504