| L(s) = 1 | + (−2.68 + 3.20i)5-s + (−4.88 − 1.77i)7-s + (4.52 + 5.38i)11-s + (3.33 − 18.8i)13-s + (−20.3 − 11.7i)17-s + (−11.7 − 20.4i)19-s + (−8.16 − 22.4i)23-s + (1.30 + 7.41i)25-s + (−21.2 + 3.74i)29-s + (24.0 − 8.73i)31-s + (18.8 − 10.8i)35-s + (−6.81 + 11.8i)37-s + (−50.7 − 8.94i)41-s + (3.55 − 2.98i)43-s + (−1.64 + 4.51i)47-s + ⋯ |
| L(s) = 1 | + (−0.537 + 0.640i)5-s + (−0.698 − 0.254i)7-s + (0.411 + 0.489i)11-s + (0.256 − 1.45i)13-s + (−1.19 − 0.692i)17-s + (−0.620 − 1.07i)19-s + (−0.355 − 0.975i)23-s + (0.0523 + 0.296i)25-s + (−0.731 + 0.128i)29-s + (0.774 − 0.281i)31-s + (0.537 − 0.310i)35-s + (−0.184 + 0.319i)37-s + (−1.23 − 0.218i)41-s + (0.0826 − 0.0693i)43-s + (−0.0349 + 0.0960i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.248560 - 0.495065i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.248560 - 0.495065i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (2.68 - 3.20i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (4.88 + 1.77i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (-4.52 - 5.38i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (-3.33 + 18.8i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (20.3 + 11.7i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (11.7 + 20.4i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (8.16 + 22.4i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (21.2 - 3.74i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (-24.0 + 8.73i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (6.81 - 11.8i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (50.7 + 8.94i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-3.55 + 2.98i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (1.64 - 4.51i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + 67.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (55.7 - 66.4i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-50.3 - 18.3i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-3.49 + 19.8i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-85.2 - 49.2i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (69.6 + 120. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-18.3 - 103. i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-79.2 + 13.9i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-58.0 + 33.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (112. - 94.0i)T + (1.63e3 - 9.26e3i)T^{2} \) |
| 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
−10.98103779022786161716706903299, −10.29285687845638730572294781414, −9.229255429555401970964423266639, −8.181437011179324471006177158151, −7.03557585269530556435967451175, −6.46485699161192623716971622747, −4.93032356424860554094193318230, −3.69429579299223271620798190395, −2.59226606130255891535964321974, −0.25253301105960345240148463239,
1.78416187247620814912290926439, 3.64835238247719929680947832483, 4.45706533132682947543640657389, 6.00662264881370328514851314405, 6.73052431706009997671229416435, 8.158064514952281735485012254411, 8.867714170168296313491998112184, 9.690372991869854536341880637581, 10.94213459563894388474464878408, 11.79920861273015479460460638213
