| L(s) = 1 | + (1.70 + 0.283i)3-s + (−1.19 + 3.27i)5-s + (0.573 + 0.101i)7-s + (2.83 + 0.968i)9-s + (−1.42 + 0.519i)11-s + (−4.76 + 3.99i)13-s + (−2.96 + 5.25i)15-s + (0.858 − 0.495i)17-s + (−3.76 − 2.17i)19-s + (0.950 + 0.335i)21-s + (0.715 + 4.05i)23-s + (−5.46 − 4.58i)25-s + (4.57 + 2.46i)27-s + (0.165 − 0.197i)29-s + (9.95 − 1.75i)31-s + ⋯ |
| L(s) = 1 | + (0.986 + 0.163i)3-s + (−0.532 + 1.46i)5-s + (0.216 + 0.0382i)7-s + (0.946 + 0.322i)9-s + (−0.430 + 0.156i)11-s + (−1.32 + 1.10i)13-s + (−0.765 + 1.35i)15-s + (0.208 − 0.120i)17-s + (−0.863 − 0.498i)19-s + (0.207 + 0.0731i)21-s + (0.149 + 0.845i)23-s + (−1.09 − 0.916i)25-s + (0.880 + 0.473i)27-s + (0.0307 − 0.0366i)29-s + (1.78 − 0.315i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.932448 + 1.41954i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.932448 + 1.41954i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.70 - 0.283i)T \) |
| good | 5 | \( 1 + (1.19 - 3.27i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.573 - 0.101i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.42 - 0.519i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (4.76 - 3.99i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.858 + 0.495i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.76 + 2.17i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.715 - 4.05i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.165 + 0.197i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-9.95 + 1.75i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (5.32 + 9.22i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.47 - 5.33i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.0229 + 0.0631i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.42 - 8.08i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 1.96iT - 53T^{2} \) |
| 59 | \( 1 + (-11.1 - 4.05i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.98 - 11.2i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.37 - 5.21i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.76 - 3.05i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.283 - 0.491i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.27 + 9.86i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.23 + 1.03i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-4.61 - 2.66i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.38 + 3.05i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.27082286644444758526623354174, −9.705150259063737012175321274151, −8.734743717069920986442374425519, −7.66767567429021856877187015631, −7.26615780384023017384195130835, −6.43504710521804461304665800414, −4.82242070791628690255152676692, −3.98407577016922550119933657713, −2.84882576224095179174727137987, −2.20495200788747492646474528446,
0.71535157546781709348622194488, 2.21049045893642219466744913120, 3.40938247580010415002117968830, 4.61421106353563757174577008738, 5.10035192393304706596334866876, 6.59277779387886945701165869285, 7.82210661760777807555126998876, 8.187303185528579440201223243999, 8.773285279578511083038473470911, 9.872395087436977109594050556120
