L(s) = 1 | + (−0.984 − 0.173i)2-s + (0.642 + 0.766i)3-s + (0.939 + 0.342i)4-s + (−0.5 − 0.866i)6-s + (−0.866 − 0.5i)8-s + (−0.173 + 0.984i)9-s + (0.342 + 0.939i)12-s + (−0.168 + 0.0451i)13-s + (0.766 + 0.642i)16-s + (0.342 − 0.939i)18-s + (−0.866 − 0.5i)23-s + (−0.173 − 0.984i)24-s + (0.866 − 0.5i)25-s + (0.173 − 0.0151i)26-s + (−0.866 + 0.500i)27-s + ⋯ |
L(s) = 1 | + (−0.984 − 0.173i)2-s + (0.642 + 0.766i)3-s + (0.939 + 0.342i)4-s + (−0.5 − 0.866i)6-s + (−0.866 − 0.5i)8-s + (−0.173 + 0.984i)9-s + (0.342 + 0.939i)12-s + (−0.168 + 0.0451i)13-s + (0.766 + 0.642i)16-s + (0.342 − 0.939i)18-s + (−0.866 − 0.5i)23-s + (−0.173 − 0.984i)24-s + (0.866 − 0.5i)25-s + (0.173 − 0.0151i)26-s + (−0.866 + 0.500i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9864919420\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9864919420\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 3 | \( 1 + (-0.642 - 0.766i)T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 13 | \( 1 + (0.168 - 0.0451i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (-1.75 - 0.469i)T + (0.866 + 0.5i)T^{2} \) |
| 31 | \( 1 + (-1.62 - 0.939i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (0.984 - 1.70i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + 0.684iT - T^{2} \) |
| 73 | \( 1 - 1.87iT - T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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
−8.812649929268727237676879216548, −8.410823335831075428396139978223, −7.86784260022991193438774756072, −6.81459101116374412826732792114, −6.23745878751392808006776207611, −4.95780035611041142303178450289, −4.27800024906354046497515352432, −3.04396024375629133360468612843, −2.66077809144382579176791954795, −1.33854489210252685753696748890,
0.810354017646453786366549538125, 1.95951150223519982539299276159, 2.72346988929557765258677465998, 3.66148024698402527953010552923, 4.98888641582207536236322495526, 6.07225129551954105019672550909, 6.61504664532325320500773969485, 7.32604426645850538823417661297, 8.057239806430977256692117680023, 8.505366528793570807037155693670