| L(s) = 1 | + (0.298 − 0.355i)5-s + (−10.1 − 3.69i)7-s + (10.2 + 12.1i)11-s + (−3.11 + 17.6i)13-s + (22.6 + 13.0i)17-s + (1.77 + 3.08i)19-s + (−1.41 − 3.87i)23-s + (4.30 + 24.4i)25-s + (−41.0 + 7.23i)29-s + (−6.62 + 2.41i)31-s + (−4.34 + 2.50i)35-s + (4.92 − 8.53i)37-s + (42.8 + 7.56i)41-s + (−27.2 + 22.8i)43-s + (5.51 − 15.1i)47-s + ⋯ |
| L(s) = 1 | + (0.0597 − 0.0711i)5-s + (−1.44 − 0.527i)7-s + (0.930 + 1.10i)11-s + (−0.239 + 1.36i)13-s + (1.33 + 0.769i)17-s + (0.0936 + 0.162i)19-s + (−0.0613 − 0.168i)23-s + (0.172 + 0.976i)25-s + (−1.41 + 0.249i)29-s + (−0.213 + 0.0777i)31-s + (−0.124 + 0.0716i)35-s + (0.133 − 0.230i)37-s + (1.04 + 0.184i)41-s + (−0.633 + 0.531i)43-s + (0.117 − 0.322i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.873754 + 0.779651i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.873754 + 0.779651i\) |
| \(L(2)\) |
|
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 | \( 1 + (-0.298 + 0.355i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (10.1 + 3.69i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (-10.2 - 12.1i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (3.11 - 17.6i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-22.6 - 13.0i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-1.77 - 3.08i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (1.41 + 3.87i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (41.0 - 7.23i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (6.62 - 2.41i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (-4.92 + 8.53i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-42.8 - 7.56i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (27.2 - 22.8i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-5.51 + 15.1i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 - 75.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-18.4 + 21.9i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (55.9 + 20.3i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (4.03 - 22.8i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-32.2 - 18.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (26.0 + 45.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (20.9 + 118. i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (115. - 20.2i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-117. + 67.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (72.3 - 60.6i)T + (1.63e3 - 9.26e3i)T^{2} \) |
| 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
−11.76956793787317214808404238568, −10.53106120370162782101629913543, −9.520695162855550431156320806113, −9.267149933254072243932780061982, −7.52853337688793743789479527136, −6.83950420598854756026279463171, −5.88277814962579853218075697704, −4.30678665027895264575510823355, −3.43119201185511821565303245219, −1.60763770762425494453572628593,
0.57045178077212769498277283382, 2.86173209660517081932401250970, 3.61467039437040704748677001343, 5.50909607752273598328557703222, 6.11072336683557659410316310836, 7.28045535648450089877848661457, 8.431433031804770964918814622244, 9.460326541033665252090910702223, 10.04301369637883511505417898974, 11.23556114046012503578047245293
