| L(s) = 1 | + (−1.73 − 0.0302i)3-s + (−0.721 + 0.860i)5-s + (−1.31 − 0.479i)7-s + (2.99 + 0.104i)9-s + (−1.05 − 1.26i)11-s + (−0.671 − 0.118i)13-s + (1.27 − 1.46i)15-s + (0.907 − 1.57i)17-s + (−2.96 + 1.71i)19-s + (2.26 + 0.870i)21-s + (3.62 − 1.31i)23-s + (0.649 + 3.68i)25-s + (−5.18 − 0.272i)27-s + (3.92 − 0.691i)29-s + (6.21 − 2.26i)31-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0174i)3-s + (−0.322 + 0.384i)5-s + (−0.498 − 0.181i)7-s + (0.999 + 0.0349i)9-s + (−0.319 − 0.380i)11-s + (−0.186 − 0.0328i)13-s + (0.329 − 0.378i)15-s + (0.220 − 0.381i)17-s + (−0.680 + 0.393i)19-s + (0.494 + 0.189i)21-s + (0.755 − 0.274i)23-s + (0.129 + 0.736i)25-s + (−0.998 − 0.0523i)27-s + (0.728 − 0.128i)29-s + (1.11 − 0.406i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.904894 - 0.0115991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.904894 - 0.0115991i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.73 + 0.0302i)T \) |
| good | 5 | \( 1 + (0.721 - 0.860i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (1.31 + 0.479i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (1.05 + 1.26i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.671 + 0.118i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.907 + 1.57i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.96 - 1.71i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.62 + 1.31i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-3.92 + 0.691i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-6.21 + 2.26i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.83 - 2.21i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.952 + 5.40i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.26 - 8.65i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-11.5 - 4.20i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 6.89iT - 53T^{2} \) |
| 59 | \( 1 + (-6.78 + 8.09i)T + (-10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.24 + 3.40i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-11.9 - 2.10i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (6.80 - 11.7i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.73 - 8.20i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.59 + 9.03i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.29 - 0.228i)T + (77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (3.52 + 6.10i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.33 - 5.31i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−10.23912856564789911889761464152, −9.607226002237155350165950478749, −8.376166080043085039173259415698, −7.40602814202882327199254769699, −6.66356584944105693905559645214, −5.88533505171454185987591154457, −4.88588662453120558359827983006, −3.89217319201308867367889506554, −2.66432277574760145168860419670, −0.78057006884272150878004904807,
0.826865552139785460870070678355, 2.55705750302516966300178598749, 4.09558395086235264758555667839, 4.82524691843895674914420498553, 5.81529895992828499712898978674, 6.63940395412959947955175536053, 7.46463063397385572381758926010, 8.514414216243318648954601167220, 9.434238226578987900378324833323, 10.34101193749871209662652710779
