L(s) = 1 | + (−0.895 + 1.55i)5-s + (−0.0213 + 2.64i)7-s + (−2.07 + 1.20i)11-s + (4.23 + 2.44i)13-s + 3.66·17-s − 3.01i·19-s + (3.26 + 1.88i)23-s + (0.897 + 1.55i)25-s + (5.68 − 3.28i)29-s + (−4.02 − 2.32i)31-s + (−4.08 − 2.40i)35-s + 9.36·37-s + (−4.04 + 6.99i)41-s + (3.48 + 6.02i)43-s + (−2.56 − 4.44i)47-s + ⋯ |
L(s) = 1 | + (−0.400 + 0.693i)5-s + (−0.00808 + 0.999i)7-s + (−0.627 + 0.362i)11-s + (1.17 + 0.678i)13-s + 0.888·17-s − 0.692i·19-s + (0.680 + 0.392i)23-s + (0.179 + 0.310i)25-s + (1.05 − 0.609i)29-s + (−0.722 − 0.417i)31-s + (−0.690 − 0.405i)35-s + 1.53·37-s + (−0.631 + 1.09i)41-s + (0.530 + 0.919i)43-s + (−0.374 − 0.648i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.578314134\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.578314134\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.0213 - 2.64i)T \) |
good | 5 | \( 1 + (0.895 - 1.55i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.07 - 1.20i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.23 - 2.44i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.66T + 17T^{2} \) |
| 19 | \( 1 + 3.01iT - 19T^{2} \) |
| 23 | \( 1 + (-3.26 - 1.88i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.68 + 3.28i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.02 + 2.32i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.36T + 37T^{2} \) |
| 41 | \( 1 + (4.04 - 6.99i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.48 - 6.02i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.56 + 4.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (7.29 - 12.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (9.81 - 5.66i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.285 + 0.493i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.96iT - 71T^{2} \) |
| 73 | \( 1 + 12.3iT - 73T^{2} \) |
| 79 | \( 1 + (-1.51 - 2.62i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.00 - 12.1i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3.74T + 89T^{2} \) |
| 97 | \( 1 + (4.77 - 2.75i)T + (48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.075915508289398380166625398231, −8.092762303266407322375475747696, −7.59543489850103304574331009372, −6.61190916856891361086670893662, −6.04486592685145438603367484172, −5.15454451084010827923071290719, −4.28409082279955825102998117243, −3.18024700535172456315055728343, −2.63670736051312096742255449403, −1.33674123627428007854371030844,
0.56498261337776937636888816072, 1.36431360993767287238165440278, 3.02054548887271034440862943680, 3.68106640231873918373422619051, 4.56809181598013541524115550516, 5.35408294007465060122010084501, 6.15538811425132321387807820094, 7.06707703765970348339445047012, 7.985802416959716906860636296565, 8.238889774058705399996934758618