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 | \( 1 - 1.19e3iT - 9.12e5T^{2} \) |
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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
−10.85364998751055096158229644504, −9.651286295969634552529118890065, −9.099851938657947808053443096713, −8.591800894186639078630778155119, −7.23309981282868927389045938349, −6.54077010408202654980242886852, −4.85660397513698427082678450176, −3.52032478365351937179677829588, −2.16072464017595473136054007757, −1.10949179434415875684113430141,
0.39926928109252492405490738615, 1.76638808693315432262294070571, 3.55854158807932015343777949481, 5.14223358155300151759586723811, 6.38180735755859735660111534015, 6.91426930939092796795416796274, 8.134542575587845168300892554899, 8.555267802511282310935954318853, 9.605167564433087778903462252296, 10.39211006765497974064010210166