| L(s) = 1 | + (−19.1 − 6.97i)3-s + (−4.90 − 27.7i)5-s + (95.7 − 165. i)7-s + (132. + 111. i)9-s + (28.9 + 50.1i)11-s + (−1.07e3 + 390. i)13-s + (−100. + 567. i)15-s + (200. − 168. i)17-s + (−516. + 1.48e3i)19-s + (−2.99e3 + 2.51e3i)21-s + (−495. + 2.80e3i)23-s + (2.18e3 − 796. i)25-s + (712. + 1.23e3i)27-s + (−4.96e3 − 4.16e3i)29-s + (−1.29e3 + 2.23e3i)31-s + ⋯ |
| L(s) = 1 | + (−1.22 − 0.447i)3-s + (−0.0876 − 0.497i)5-s + (0.738 − 1.27i)7-s + (0.545 + 0.458i)9-s + (0.0721 + 0.124i)11-s + (−1.75 + 0.640i)13-s + (−0.114 + 0.650i)15-s + (0.168 − 0.141i)17-s + (−0.328 + 0.944i)19-s + (−1.48 + 1.24i)21-s + (−0.195 + 1.10i)23-s + (0.700 − 0.254i)25-s + (0.187 + 0.325i)27-s + (−1.09 − 0.919i)29-s + (−0.241 + 0.418i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0215459 + 0.0461679i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0215459 + 0.0461679i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (516. - 1.48e3i)T \) |
| good | 3 | \( 1 + (19.1 + 6.97i)T + (186. + 156. i)T^{2} \) |
| 5 | \( 1 + (4.90 + 27.7i)T + (-2.93e3 + 1.06e3i)T^{2} \) |
| 7 | \( 1 + (-95.7 + 165. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-28.9 - 50.1i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (1.07e3 - 390. i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-200. + 168. i)T + (2.46e5 - 1.39e6i)T^{2} \) |
| 23 | \( 1 + (495. - 2.80e3i)T + (-6.04e6 - 2.20e6i)T^{2} \) |
| 29 | \( 1 + (4.96e3 + 4.16e3i)T + (3.56e6 + 2.01e7i)T^{2} \) |
| 31 | \( 1 + (1.29e3 - 2.23e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 3.34e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.47e4 + 5.38e3i)T + (8.87e7 + 7.44e7i)T^{2} \) |
| 43 | \( 1 + (-1.19e3 - 6.79e3i)T + (-1.38e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (-5.05e3 - 4.23e3i)T + (3.98e7 + 2.25e8i)T^{2} \) |
| 53 | \( 1 + (5.49e3 - 3.11e4i)T + (-3.92e8 - 1.43e8i)T^{2} \) |
| 59 | \( 1 + (-2.46e4 + 2.06e4i)T + (1.24e8 - 7.04e8i)T^{2} \) |
| 61 | \( 1 + (-1.18e3 + 6.71e3i)T + (-7.93e8 - 2.88e8i)T^{2} \) |
| 67 | \( 1 + (2.41e4 + 2.02e4i)T + (2.34e8 + 1.32e9i)T^{2} \) |
| 71 | \( 1 + (5.01e3 + 2.84e4i)T + (-1.69e9 + 6.17e8i)T^{2} \) |
| 73 | \( 1 + (3.54e3 + 1.29e3i)T + (1.58e9 + 1.33e9i)T^{2} \) |
| 79 | \( 1 + (-3.77e3 - 1.37e3i)T + (2.35e9 + 1.97e9i)T^{2} \) |
| 83 | \( 1 + (3.50e4 - 6.06e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (1.04e5 - 3.79e4i)T + (4.27e9 - 3.58e9i)T^{2} \) |
| 97 | \( 1 + (-2.24e3 + 1.88e3i)T + (1.49e9 - 8.45e9i)T^{2} \) |
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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
−12.53466505918865990574533514932, −11.81131320003382363124071960781, −10.81659606561955356939173585087, −9.654057143586339564336859855228, −7.78860193681259463165159327878, −6.93400917867505880100372546687, −5.37738245497578956357957094081, −4.32396351578212455007419397802, −1.47088616310441391138951724315, −0.02611567778689995599487679185,
2.52682767028071057130103622800, 4.84020169645332306320990274308, 5.57091539491914009765333940101, 7.01859352846345595379144395966, 8.594973364016375189653257773020, 10.03392030945685907718738116881, 11.05870217153146824998265983188, 11.85129624737563699351010338815, 12.73222212164019762753847041448, 14.75389198692059686686070586069
