| L(s) = 1 | + (0.684 + 1.23i)2-s + (0.464 + 1.66i)3-s + (−1.06 + 1.69i)4-s + (−0.295 − 0.384i)5-s + (−1.74 + 1.71i)6-s + (−0.264 − 0.0707i)7-s + (−2.82 − 0.152i)8-s + (−2.56 + 1.55i)9-s + (0.273 − 0.628i)10-s + (1.17 − 0.154i)11-s + (−3.32 − 0.984i)12-s + (−0.871 + 6.61i)13-s + (−0.0933 − 0.375i)14-s + (0.504 − 0.671i)15-s + (−1.74 − 3.59i)16-s + 6.96·17-s + ⋯ |
| L(s) = 1 | + (0.484 + 0.874i)2-s + (0.268 + 0.963i)3-s + (−0.530 + 0.847i)4-s + (−0.131 − 0.171i)5-s + (−0.712 + 0.701i)6-s + (−0.0997 − 0.0267i)7-s + (−0.998 − 0.0539i)8-s + (−0.856 + 0.516i)9-s + (0.0865 − 0.198i)10-s + (0.354 − 0.0466i)11-s + (−0.958 − 0.284i)12-s + (−0.241 + 1.83i)13-s + (−0.0249 − 0.100i)14-s + (0.130 − 0.173i)15-s + (−0.436 − 0.899i)16-s + 1.68·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.367752 + 1.48648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.367752 + 1.48648i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.684 - 1.23i)T \) |
| 3 | \( 1 + (-0.464 - 1.66i)T \) |
| good | 5 | \( 1 + (0.295 + 0.384i)T + (-1.29 + 4.82i)T^{2} \) |
| 7 | \( 1 + (0.264 + 0.0707i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.17 + 0.154i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (0.871 - 6.61i)T + (-12.5 - 3.36i)T^{2} \) |
| 17 | \( 1 - 6.96T + 17T^{2} \) |
| 19 | \( 1 + (2.68 + 6.48i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.562 - 2.09i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-5.93 - 4.55i)T + (7.50 + 28.0i)T^{2} \) |
| 31 | \( 1 + (-2.80 - 1.61i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.576 + 0.238i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-3.81 + 1.02i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.785 - 5.96i)T + (-41.5 + 11.1i)T^{2} \) |
| 47 | \( 1 + (-0.925 + 0.534i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.57 - 1.89i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-5.98 + 4.59i)T + (15.2 - 56.9i)T^{2} \) |
| 61 | \( 1 + (3.88 - 5.06i)T + (-15.7 - 58.9i)T^{2} \) |
| 67 | \( 1 + (0.803 - 6.10i)T + (-64.7 - 17.3i)T^{2} \) |
| 71 | \( 1 + (5.64 + 5.64i)T + 71iT^{2} \) |
| 73 | \( 1 + (1.52 - 1.52i)T - 73iT^{2} \) |
| 79 | \( 1 + (6.66 + 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.70 + 6.67i)T + (21.4 + 80.1i)T^{2} \) |
| 89 | \( 1 + (-9.28 + 9.28i)T - 89iT^{2} \) |
| 97 | \( 1 + (-2.92 - 5.07i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.16101234974999844963764449469, −11.48173492354064299004705151305, −10.08858049956131290746334978000, −9.125739179268928259506867278094, −8.535794032062690542278584689333, −7.25291617320525595936853382950, −6.23853240491648556228971472149, −4.90643526057785799842494528768, −4.25209891643216139362637587049, −2.97688425751675381671547275953,
1.06118507847594203691803007497, 2.71523329499202048512736146134, 3.64931267048716395469521471852, 5.42939530216475625731791209950, 6.19686119651950147020477468871, 7.67936390865672682075814980629, 8.435759130300227533638963785433, 9.837227799727167532743567695215, 10.50561310260769976143746962657, 11.74140877639250727345316435481
