L(s) = 1 | + (1.04 + 0.948i)2-s + (0.201 + 1.98i)4-s + (1.39 + 3.36i)5-s + (−1.05 − 1.05i)7-s + (−1.67 + 2.27i)8-s + (−1.73 + 4.85i)10-s + (−1.50 − 3.64i)11-s + (−2.23 − 0.927i)13-s + (−0.106 − 2.11i)14-s + (−3.91 + 0.803i)16-s + 7.49·17-s + (0.818 − 1.97i)19-s + (−6.42 + 3.45i)20-s + (1.87 − 5.25i)22-s + (5.80 + 5.80i)23-s + ⋯ |
L(s) = 1 | + (0.741 + 0.670i)2-s + (0.100 + 0.994i)4-s + (0.624 + 1.50i)5-s + (−0.399 − 0.399i)7-s + (−0.592 + 0.805i)8-s + (−0.547 + 1.53i)10-s + (−0.455 − 1.09i)11-s + (−0.620 − 0.257i)13-s + (−0.0285 − 0.564i)14-s + (−0.979 + 0.200i)16-s + 1.81·17-s + (0.187 − 0.453i)19-s + (−1.43 + 0.773i)20-s + (0.399 − 1.12i)22-s + (1.21 + 1.21i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19106 + 1.49659i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19106 + 1.49659i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.04 - 0.948i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.39 - 3.36i)T + (-3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (1.05 + 1.05i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.50 + 3.64i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (2.23 + 0.927i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 7.49T + 17T^{2} \) |
| 19 | \( 1 + (-0.818 + 1.97i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-5.80 - 5.80i)T + 23iT^{2} \) |
| 29 | \( 1 + (0.326 + 0.135i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 1.71iT - 31T^{2} \) |
| 37 | \( 1 + (-0.387 + 0.160i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-1.50 + 1.50i)T - 41iT^{2} \) |
| 43 | \( 1 + (-7.40 + 3.06i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 7.27iT - 47T^{2} \) |
| 53 | \( 1 + (3.94 - 1.63i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (12.7 - 5.30i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (0.579 - 1.40i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (7.96 + 3.29i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (4.75 - 4.75i)T - 71iT^{2} \) |
| 73 | \( 1 + (7.99 + 7.99i)T + 73iT^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + (1.35 + 0.560i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-4.75 - 4.75i)T + 89iT^{2} \) |
| 97 | \( 1 + 1.28T + 97T^{2} \) |
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
−12.18800927081291482862191694725, −11.13228473629178043971154799216, −10.35474779305683188090397442332, −9.273777817368261237353256493279, −7.69758774772426776938013189011, −7.18202006476356336137317570179, −6.04855015720449351136572102955, −5.35538048344359973259167116780, −3.44309579463704637012814315904, −2.87272533525373124301207978604,
1.35063564988639649901028759499, 2.79260687453899908960236729392, 4.53797973329973152379896609991, 5.17866411966229962437868757242, 6.14132669396933427933985531129, 7.67240052027523606508288725276, 9.161051396177110023053596779701, 9.645829280228284534702828814863, 10.53037575898101676587862757615, 12.05843394587826102442549183747