| L(s) = 1 | + (−0.104 + 0.994i)2-s + (0.752 − 1.55i)3-s + (−0.978 − 0.207i)4-s + (−0.112 + 0.251i)5-s + (1.47 + 0.911i)6-s + (−2.57 − 0.597i)7-s + (0.309 − 0.951i)8-s + (−1.86 − 2.34i)9-s + (−0.238 − 0.137i)10-s + (−0.877 − 3.19i)11-s + (−1.06 + 1.36i)12-s + (0.376 − 0.518i)13-s + (0.863 − 2.50i)14-s + (0.308 + 0.364i)15-s + (0.913 + 0.406i)16-s + (−0.638 − 6.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (0.434 − 0.900i)3-s + (−0.489 − 0.103i)4-s + (−0.0500 + 0.112i)5-s + (0.601 + 0.372i)6-s + (−0.974 − 0.225i)7-s + (0.109 − 0.336i)8-s + (−0.622 − 0.782i)9-s + (−0.0754 − 0.0435i)10-s + (−0.264 − 0.964i)11-s + (−0.306 + 0.395i)12-s + (0.104 − 0.143i)13-s + (0.230 − 0.668i)14-s + (0.0795 + 0.0940i)15-s + (0.228 + 0.101i)16-s + (−0.154 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0189 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0189 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.679829 - 0.692807i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.679829 - 0.692807i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (-0.752 + 1.55i)T \) |
| 7 | \( 1 + (2.57 + 0.597i)T \) |
| 11 | \( 1 + (0.877 + 3.19i)T \) |
| good | 5 | \( 1 + (0.112 - 0.251i)T + (-3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (-0.376 + 0.518i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.638 + 6.07i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.228 - 1.07i)T + (-17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (1.05 - 0.608i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.73 + 8.40i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (4.57 - 2.03i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-4.83 + 5.37i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (-0.939 + 2.89i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 1.12iT - 43T^{2} \) |
| 47 | \( 1 + (-0.496 - 2.33i)T + (-42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-3.33 - 7.48i)T + (-35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-1.83 + 8.65i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (1.80 - 4.04i)T + (-40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (5.31 - 9.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.62 - 9.11i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.37 + 11.1i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-12.3 - 1.29i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-3.41 + 2.48i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-15.7 + 9.07i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.14 + 5.91i)T + (29.9 + 92.2i)T^{2} \) |
| 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
−10.83868995740123043633161783018, −9.513121076457162899830083120481, −9.000920446476428384704050684407, −7.83288150344877028001220561364, −7.23030987909618188226076422024, −6.29139663604935107782314875282, −5.50746206687524097673275482289, −3.75921403421324699861266441061, −2.71461199038171765089402116076, −0.56983322169234017277840333418,
2.14777850652894654874600245164, 3.33777869959822510245663344416, 4.23801377596937083566088088160, 5.29949777736294274193359745815, 6.59421862766793657790590592134, 7.981239410826950782473244018598, 8.900414598259288669220039911316, 9.577711731789670762918117088499, 10.36119055119418038982320292831, 10.95749477800489886397088455739
