| L(s) = 1 | + (5.14 − 26.5i)3-s − 71.4i·5-s − 129.·7-s + (−676. − 272. i)9-s − 1.19e3i·11-s − 3.40e3·13-s + (−1.89e3 − 367. i)15-s − 6.56e3i·17-s + 1.96e3·19-s + (−666. + 3.43e3i)21-s + 2.01e4i·23-s + 1.05e4·25-s + (−1.07e4 + 1.65e4i)27-s − 2.02e4i·29-s − 2.45e4·31-s + ⋯ |
| L(s) = 1 | + (0.190 − 0.981i)3-s − 0.571i·5-s − 0.377·7-s + (−0.927 − 0.373i)9-s − 0.899i·11-s − 1.55·13-s + (−0.561 − 0.108i)15-s − 1.33i·17-s + 0.286·19-s + (−0.0719 + 0.371i)21-s + 1.65i·23-s + 0.673·25-s + (−0.543 + 0.839i)27-s − 0.829i·29-s − 0.824·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1111713085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1111713085\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-5.14 + 26.5i)T \) |
| 7 | \( 1 + 129.T \) |
| good | 5 | \( 1 + 71.4iT - 1.56e4T^{2} \) |
| 11 | \( 1 + 1.19e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.40e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 6.56e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.96e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 2.01e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.02e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.45e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 6.47e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 5.51e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.28e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.34e3iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 9.92e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.56e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.19e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.56e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 5.89e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 6.79e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 9.00e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 2.83e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 3.98e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 4.44e5T + 8.32e11T^{2} \) |
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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
−9.446294865167096837483428904163, −8.996473280359649050341157264940, −7.64336436759594254940661855933, −7.19882269749921500109604130901, −5.85767910911846770316262190355, −5.03144023146443846347825540246, −3.35964950737822326718691641695, −2.35155977091318413068308678781, −0.954139923721548184339924626484, −0.02828465259906207283414629728,
2.14802715150382595072025311102, 3.12970079283092282623125189333, 4.30774324294723111267354945512, 5.18096747205609691060157593538, 6.49667658439350044333789605341, 7.44613684927232887772060370695, 8.633036045464276160627219462980, 9.575130975330899544861445024645, 10.34150433210045292316342769348, 10.85794516713175239545968364868
