| L(s) = 1 | + (−0.0410 + 0.0224i)3-s + (−0.288 + 2.21i)5-s + (1.73 + 2.31i)7-s + (−1.62 + 2.52i)9-s + (0.565 − 0.258i)11-s + (−0.890 − 0.666i)13-s + (−0.0378 − 0.0974i)15-s + (−0.149 − 2.08i)17-s + (1.43 + 1.66i)19-s + (−0.122 − 0.0560i)21-s + (−4.23 + 2.25i)23-s + (−4.83 − 1.28i)25-s + (0.0200 − 0.279i)27-s + (−1.86 − 1.61i)29-s + (−8.11 + 2.38i)31-s + ⋯ |
| L(s) = 1 | + (−0.0236 + 0.0129i)3-s + (−0.129 + 0.991i)5-s + (0.654 + 0.873i)7-s + (−0.540 + 0.840i)9-s + (0.170 − 0.0778i)11-s + (−0.246 − 0.184i)13-s + (−0.00976 − 0.0251i)15-s + (−0.0361 − 0.505i)17-s + (0.330 + 0.381i)19-s + (−0.0267 − 0.0122i)21-s + (−0.882 + 0.469i)23-s + (−0.966 − 0.256i)25-s + (0.00384 − 0.0538i)27-s + (−0.347 − 0.300i)29-s + (−1.45 + 0.427i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.633 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.511172 + 1.07839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.511172 + 1.07839i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.288 - 2.21i)T \) |
| 23 | \( 1 + (4.23 - 2.25i)T \) |
| good | 3 | \( 1 + (0.0410 - 0.0224i)T + (1.62 - 2.52i)T^{2} \) |
| 7 | \( 1 + (-1.73 - 2.31i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (-0.565 + 0.258i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.890 + 0.666i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (0.149 + 2.08i)T + (-16.8 + 2.41i)T^{2} \) |
| 19 | \( 1 + (-1.43 - 1.66i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (1.86 + 1.61i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (8.11 - 2.38i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (0.876 - 0.190i)T + (33.6 - 15.3i)T^{2} \) |
| 41 | \( 1 + (-6.19 + 3.98i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-2.73 - 5.01i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (-1.83 - 1.83i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.56 + 1.91i)T + (14.9 - 50.8i)T^{2} \) |
| 59 | \( 1 + (6.13 - 0.881i)T + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-3.88 - 13.2i)T + (-51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-4.76 - 12.7i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (0.404 - 0.884i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-3.80 - 0.272i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (-0.310 - 2.15i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-2.23 - 10.2i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (9.66 + 2.83i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-2.33 + 10.7i)T + (-88.2 - 40.2i)T^{2} \) |
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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
−10.48950796864858107160392987981, −9.586125066792035453358634425059, −8.633053135139509935470334450941, −7.79043193694633177083381897788, −7.17341486942051262908320452365, −5.83102931863586784974898061064, −5.40589124976117085334800088371, −4.06248280053956067279299212164, −2.83414680197457767339290481977, −2.00523153455043365024668070628,
0.55241513355585332235054251681, 1.86907327357860112653865130813, 3.62292156312380217528291606347, 4.35234247392192843087204682927, 5.33089746819251760873042116428, 6.27871534753680256114864259518, 7.39721088737483382484158662249, 8.094431999341647171264307668522, 9.019437996459010877378702280681, 9.570623365337690344380296964899
