| L(s) = 1 | + (−17.1 + 12.4i)3-s + (26.4 + 81.3i)5-s + (174. + 126. i)7-s + (63.6 − 195. i)9-s + (157. − 369. i)11-s + (−162. + 500. i)13-s + (−1.46e3 − 1.06e3i)15-s + (612. + 1.88e3i)17-s + (−702. + 510. i)19-s − 4.56e3·21-s + 104.·23-s + (−3.38e3 + 2.45e3i)25-s + (−242. − 747. i)27-s + (−3.48e3 − 2.53e3i)29-s + (2.45e3 − 7.56e3i)31-s + ⋯ |
| L(s) = 1 | + (−1.09 + 0.798i)3-s + (0.472 + 1.45i)5-s + (1.34 + 0.977i)7-s + (0.261 − 0.805i)9-s + (0.391 − 0.920i)11-s + (−0.266 + 0.821i)13-s + (−1.68 − 1.22i)15-s + (0.514 + 1.58i)17-s + (−0.446 + 0.324i)19-s − 2.25·21-s + 0.0412·23-s + (−1.08 + 0.787i)25-s + (−0.0640 − 0.197i)27-s + (−0.769 − 0.559i)29-s + (0.459 − 1.41i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.244220 + 1.35579i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.244220 + 1.35579i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (-157. + 369. i)T \) |
| good | 3 | \( 1 + (17.1 - 12.4i)T + (75.0 - 231. i)T^{2} \) |
| 5 | \( 1 + (-26.4 - 81.3i)T + (-2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-174. - 126. i)T + (5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (162. - 500. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-612. - 1.88e3i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (702. - 510. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 - 104.T + 6.43e6T^{2} \) |
| 29 | \( 1 + (3.48e3 + 2.53e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-2.45e3 + 7.56e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (3.02e3 + 2.19e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.20e4 + 8.78e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.13e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.08e3 - 789. i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (9.26e3 - 2.85e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.07e4 + 2.23e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (6.12e3 + 1.88e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 - 6.00e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-1.04e4 - 3.22e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-2.82e4 - 2.05e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-8.99e3 + 2.76e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-1.56e4 - 4.80e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 - 2.74e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-9.35e3 + 2.87e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−14.07452921309487991466432469357, −12.15440716106304211819967514755, −11.09705266346948947019686407539, −10.88461412395415287514952237054, −9.544518050826485122837290190317, −8.048962057085840624224627599466, −6.24062543709435567640690437412, −5.65390788459897224907968000882, −4.06164636955158419352241578359, −2.09407184853642468418798821107,
0.70402673189115091011859086988, 1.50975855226246184934656619022, 4.75442682042674051450762980083, 5.25695511620034681415308137387, 6.96415486953158721375178349351, 7.945116233907869898780844223278, 9.403248829140765166897338952615, 10.77343642295019580125184246833, 11.87263352660294740821217362250, 12.57188156974150300507685793892
