| L(s) = 1 | + (11.0 + 11.0i)3-s + (−122. − 122. i)5-s + 392.·7-s + 242. i·9-s + (−469. + 469. i)11-s + (−284. + 284. i)13-s − 2.69e3i·15-s − 4.54e3·17-s + (−2.14e3 − 2.14e3i)19-s + (4.32e3 + 4.32e3i)21-s + 2.34e4·23-s + 1.43e4i·25-s + (−2.67e3 + 2.67e3i)27-s + (−2.97e4 + 2.97e4i)29-s − 6.95e3i·31-s + ⋯ |
| L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.978 − 0.978i)5-s + 1.14·7-s + 0.333i·9-s + (−0.352 + 0.352i)11-s + (−0.129 + 0.129i)13-s − 0.799i·15-s − 0.924·17-s + (−0.312 − 0.312i)19-s + (0.466 + 0.466i)21-s + 1.92·23-s + 0.915i·25-s + (−0.136 + 0.136i)27-s + (−1.22 + 1.22i)29-s − 0.233i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 - 0.941i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.338 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.691211999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.691211999\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-11.0 - 11.0i)T \) |
| good | 5 | \( 1 + (122. + 122. i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 - 392.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (469. - 469. i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (284. - 284. i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + 4.54e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (2.14e3 + 2.14e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 - 2.34e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (2.97e4 - 2.97e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + 6.95e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (4.28e4 + 4.28e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 5.10e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-5.65e4 + 5.65e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 - 6.21e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (-1.75e5 - 1.75e5i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (1.47e5 - 1.47e5i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (-1.89e5 + 1.89e5i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (5.92e4 + 5.92e4i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 5.75e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 3.77e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 6.05e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-6.44e5 - 6.44e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 6.51e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 4.63e5T + 8.32e11T^{2} \) |
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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
−10.84728506912537994945903739467, −9.191670181720408856672361905652, −8.783124468462101192075614083014, −7.85125725351612171898416147330, −7.08907541648198388913207789111, −5.18184125237509601510303595356, −4.71350634473529515327477931766, −3.76159895243845039742212976451, −2.25221296891132985017784285717, −0.972496278137271201298636011579,
0.41828744527138787752526271211, 1.92893098292223006321306542007, 3.01072568654432825035677874217, 4.05787213717123331714206013335, 5.23606470122006243212962307260, 6.67978019143652159204059519555, 7.42934032167086290962806929502, 8.152826347766499545804062039008, 8.960210115568760914787023115768, 10.41800636429133547466585485962
