L(s) = 1 | + (−1.69 + 0.346i)3-s + (−1.79 − 3.10i)5-s + (1.83 + 1.90i)7-s + (2.75 − 1.17i)9-s + (−0.200 − 0.115i)11-s + (−1.16 + 0.673i)13-s + (4.11 + 4.64i)15-s − 7.94·17-s − 3.06i·19-s + (−3.77 − 2.60i)21-s + (−4.87 + 2.81i)23-s + (−3.91 + 6.78i)25-s + (−4.27 + 2.95i)27-s + (−2.33 − 1.34i)29-s + (−1.85 + 1.07i)31-s + ⋯ |
L(s) = 1 | + (−0.979 + 0.200i)3-s + (−0.801 − 1.38i)5-s + (0.692 + 0.720i)7-s + (0.919 − 0.392i)9-s + (−0.0604 − 0.0348i)11-s + (−0.323 + 0.186i)13-s + (1.06 + 1.19i)15-s − 1.92·17-s − 0.702i·19-s + (−0.823 − 0.567i)21-s + (−1.01 + 0.586i)23-s + (−0.783 + 1.35i)25-s + (−0.822 + 0.568i)27-s + (−0.433 − 0.250i)29-s + (−0.332 + 0.192i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00304638 + 0.115378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00304638 + 0.115378i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.69 - 0.346i)T \) |
| 7 | \( 1 + (-1.83 - 1.90i)T \) |
good | 5 | \( 1 + (1.79 + 3.10i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.200 + 0.115i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.16 - 0.673i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 7.94T + 17T^{2} \) |
| 19 | \( 1 + 3.06iT - 19T^{2} \) |
| 23 | \( 1 + (4.87 - 2.81i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.33 + 1.34i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.85 - 1.07i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.27T + 37T^{2} \) |
| 41 | \( 1 + (-0.813 - 1.40i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.927 - 1.60i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.0396 + 0.0686i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 11.6iT - 53T^{2} \) |
| 59 | \( 1 + (-6.48 - 11.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.729 + 0.420i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.05 + 8.75i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.47iT - 71T^{2} \) |
| 73 | \( 1 + 7.97iT - 73T^{2} \) |
| 79 | \( 1 + (3.30 - 5.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.41 - 11.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 3.56T + 89T^{2} \) |
| 97 | \( 1 + (-13.1 - 7.58i)T + (48.5 + 84.0i)T^{2} \) |
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
−10.74180880577826946587698161814, −9.379430024631468350178917302014, −8.786860464964429954553327237587, −7.85185826689645782795945212133, −6.71669417290161070521732872947, −5.47451105588744307233285806411, −4.79226562190621776628022534788, −4.07860659463245135252451330657, −1.83601663510186636325080478152, −0.07500115607500055236975421163,
2.11001591036628079900160255814, 3.81370750921935510630726849183, 4.61261916369425751700331074543, 5.99246402727987311059012133332, 6.98163109088236428730859128121, 7.39136292229441590966213854157, 8.446636086189232520715720904399, 10.11287881451109666323858777723, 10.67577443342268458104911897532, 11.25801010294659721349352944289