| L(s) = 1 | + (0.149 + 0.845i)3-s + (−73.5 + 61.6i)5-s + (18.6 − 32.2i)7-s + (227. − 82.8i)9-s + (−225. − 390. i)11-s + (144. − 819. i)13-s + (−63.1 − 52.9i)15-s + (553. + 201. i)17-s + (−1.56e3 − 115. i)19-s + (30.0 + 10.9i)21-s + (−1.55e3 − 1.30e3i)23-s + (1.05e3 − 5.99e3i)25-s + (208. + 360. i)27-s + (475. − 173. i)29-s + (4.46e3 − 7.74e3i)31-s + ⋯ |
| L(s) = 1 | + (0.00956 + 0.0542i)3-s + (−1.31 + 1.10i)5-s + (0.143 − 0.248i)7-s + (0.936 − 0.340i)9-s + (−0.561 − 0.971i)11-s + (0.237 − 1.34i)13-s + (−0.0724 − 0.0607i)15-s + (0.464 + 0.169i)17-s + (−0.997 − 0.0735i)19-s + (0.0148 + 0.00541i)21-s + (−0.612 − 0.513i)23-s + (0.338 − 1.91i)25-s + (0.0550 + 0.0952i)27-s + (0.105 − 0.0382i)29-s + (0.835 − 1.44i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.570767 - 0.638944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.570767 - 0.638944i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.56e3 + 115. i)T \) |
| good | 3 | \( 1 + (-0.149 - 0.845i)T + (-228. + 83.1i)T^{2} \) |
| 5 | \( 1 + (73.5 - 61.6i)T + (542. - 3.07e3i)T^{2} \) |
| 7 | \( 1 + (-18.6 + 32.2i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (225. + 390. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-144. + 819. i)T + (-3.48e5 - 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-553. - 201. i)T + (1.08e6 + 9.12e5i)T^{2} \) |
| 23 | \( 1 + (1.55e3 + 1.30e3i)T + (1.11e6 + 6.33e6i)T^{2} \) |
| 29 | \( 1 + (-475. + 173. i)T + (1.57e7 - 1.31e7i)T^{2} \) |
| 31 | \( 1 + (-4.46e3 + 7.74e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 1.18e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-1.44e3 - 8.17e3i)T + (-1.08e8 + 3.96e7i)T^{2} \) |
| 43 | \( 1 + (-9.33e3 + 7.83e3i)T + (2.55e7 - 1.44e8i)T^{2} \) |
| 47 | \( 1 + (5.82e3 - 2.11e3i)T + (1.75e8 - 1.47e8i)T^{2} \) |
| 53 | \( 1 + (2.89e3 + 2.42e3i)T + (7.26e7 + 4.11e8i)T^{2} \) |
| 59 | \( 1 + (-1.09e3 - 400. i)T + (5.47e8 + 4.59e8i)T^{2} \) |
| 61 | \( 1 + (3.95e4 + 3.32e4i)T + (1.46e8 + 8.31e8i)T^{2} \) |
| 67 | \( 1 + (1.03e4 - 3.76e3i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (5.70e4 - 4.78e4i)T + (3.13e8 - 1.77e9i)T^{2} \) |
| 73 | \( 1 + (-1.11e4 - 6.30e4i)T + (-1.94e9 + 7.09e8i)T^{2} \) |
| 79 | \( 1 + (1.28e4 + 7.29e4i)T + (-2.89e9 + 1.05e9i)T^{2} \) |
| 83 | \( 1 + (-5.01e4 + 8.68e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-1.05e4 + 5.96e4i)T + (-5.24e9 - 1.90e9i)T^{2} \) |
| 97 | \( 1 + (3.15e4 + 1.14e4i)T + (6.57e9 + 5.51e9i)T^{2} \) |
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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
−13.18064581367856535949810899219, −12.06789513473205761838029401946, −10.83712784196697377146366022053, −10.30438073430508276784741607984, −8.265681737747923616397122845676, −7.52655747660656439758982002575, −6.17906087984025952357635573142, −4.17055542746209417427259551126, −3.05551838913169758527448752649, −0.38447932515301869595767464654,
1.61709298092627933552945453838, 4.09612567034420664371454484179, 4.90462569417197982600769977563, 7.03103505498005267902191119990, 8.049740577426295422996762725267, 9.121532685343213088532303273123, 10.52800533622708580318349717960, 12.03875994692115413912360067910, 12.39378086590786932077486705274, 13.66892960000178529427196162209
