| L(s) = 1 | + (−0.308 − 1.38i)2-s + (−1.23 + 1.21i)3-s + (−1.80 + 0.851i)4-s + (−2.46 + 0.215i)5-s + (2.06 + 1.32i)6-s + (3.54 + 1.28i)7-s + (1.73 + 2.23i)8-s + (0.0321 − 2.99i)9-s + (1.05 + 3.34i)10-s + (−2.05 − 0.179i)11-s + (1.19 − 3.25i)12-s + (−1.75 − 2.51i)13-s + (0.686 − 5.28i)14-s + (2.77 − 3.27i)15-s + (2.54 − 3.08i)16-s + (−1.52 − 0.879i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.975i)2-s + (−0.710 + 0.703i)3-s + (−0.904 + 0.425i)4-s + (−1.10 + 0.0965i)5-s + (0.841 + 0.540i)6-s + (1.33 + 0.487i)7-s + (0.612 + 0.790i)8-s + (0.0107 − 0.999i)9-s + (0.335 + 1.05i)10-s + (−0.618 − 0.0540i)11-s + (0.343 − 0.939i)12-s + (−0.487 − 0.696i)13-s + (0.183 − 1.41i)14-s + (0.716 − 0.845i)15-s + (0.637 − 0.770i)16-s + (−0.369 − 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.119015 - 0.373872i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.119015 - 0.373872i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.308 + 1.38i)T \) |
| 3 | \( 1 + (1.23 - 1.21i)T \) |
| good | 5 | \( 1 + (2.46 - 0.215i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-3.54 - 1.28i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (2.05 + 0.179i)T + (10.8 + 1.91i)T^{2} \) |
| 13 | \( 1 + (1.75 + 2.51i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (1.52 + 0.879i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.557 + 2.08i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (2.89 + 7.96i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (3.62 - 5.17i)T + (-9.91 - 27.2i)T^{2} \) |
| 31 | \( 1 + (2.30 + 6.33i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-6.28 + 1.68i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.905 + 5.13i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.568 + 6.49i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-0.285 - 0.104i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-3.58 + 3.58i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.514 + 5.87i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (9.28 - 4.32i)T + (39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (0.803 - 0.562i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-13.0 - 7.52i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (9.70 - 5.60i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.6 + 1.88i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (13.5 + 9.48i)T + (28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (3.56 + 6.17i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.5 - 8.84i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−10.97402240525138988697243347054, −10.25893746858156385749811812920, −9.058934787142478798764529322161, −8.238803338665442772002060391455, −7.42217132067031838250447145148, −5.52153341099307751962236848463, −4.73363998285863134728776775930, −3.92276795082062579492289494656, −2.47664287332563064529851201616, −0.31005167776132315040637195151,
1.51134990926884581437203110614, 4.15409100641957713656726121645, 4.89262486153619906706016509839, 5.91440872579561064780808915085, 7.18402301032974031522351242165, 7.79967270733089348471706118900, 8.133848424171565275281453966200, 9.607390030713382261134686686371, 10.83698478183835124276885251346, 11.46955544914460991820000720181
