| L(s) = 1 | + (−49.6 − 8.75i)3-s + (−5.51 − 4.63i)5-s + (269. + 466. i)7-s + (1.70e3 + 619. i)9-s + (767. − 1.32e3i)11-s + (−2.55e3 + 451. i)13-s + (233. + 278. i)15-s + (−6.66e3 + 2.42e3i)17-s + (6.85e3 + 114. i)19-s + (−9.28e3 − 2.55e4i)21-s + (−9.97e3 + 8.37e3i)23-s + (−2.70e3 − 1.53e4i)25-s + (−4.72e4 − 2.72e4i)27-s + (1.04e3 − 2.86e3i)29-s + (3.53e4 − 2.04e4i)31-s + ⋯ |
| L(s) = 1 | + (−1.83 − 0.324i)3-s + (−0.0441 − 0.0370i)5-s + (0.785 + 1.36i)7-s + (2.33 + 0.850i)9-s + (0.576 − 0.998i)11-s + (−1.16 + 0.205i)13-s + (0.0691 + 0.0824i)15-s + (−1.35 + 0.493i)17-s + (0.999 + 0.0167i)19-s + (−1.00 − 2.75i)21-s + (−0.820 + 0.688i)23-s + (−0.173 − 0.981i)25-s + (−2.40 − 1.38i)27-s + (0.0427 − 0.117i)29-s + (1.18 − 0.684i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.119080 - 0.280942i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.119080 - 0.280942i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-6.85e3 - 114. i)T \) |
| good | 3 | \( 1 + (49.6 + 8.75i)T + (685. + 249. i)T^{2} \) |
| 5 | \( 1 + (5.51 + 4.63i)T + (2.71e3 + 1.53e4i)T^{2} \) |
| 7 | \( 1 + (-269. - 466. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-767. + 1.32e3i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (2.55e3 - 451. i)T + (4.53e6 - 1.65e6i)T^{2} \) |
| 17 | \( 1 + (6.66e3 - 2.42e3i)T + (1.84e7 - 1.55e7i)T^{2} \) |
| 23 | \( 1 + (9.97e3 - 8.37e3i)T + (2.57e7 - 1.45e8i)T^{2} \) |
| 29 | \( 1 + (-1.04e3 + 2.86e3i)T + (-4.55e8 - 3.82e8i)T^{2} \) |
| 31 | \( 1 + (-3.53e4 + 2.04e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 3.35e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (7.14e4 + 1.25e4i)T + (4.46e9 + 1.62e9i)T^{2} \) |
| 43 | \( 1 + (-8.90e3 - 7.47e3i)T + (1.09e9 + 6.22e9i)T^{2} \) |
| 47 | \( 1 + (8.26e4 + 3.01e4i)T + (8.25e9 + 6.92e9i)T^{2} \) |
| 53 | \( 1 + (5.96e4 + 7.10e4i)T + (-3.84e9 + 2.18e10i)T^{2} \) |
| 59 | \( 1 + (1.34e5 + 3.70e5i)T + (-3.23e10 + 2.71e10i)T^{2} \) |
| 61 | \( 1 + (1.12e5 - 9.46e4i)T + (8.94e9 - 5.07e10i)T^{2} \) |
| 67 | \( 1 + (-1.32e5 + 3.64e5i)T + (-6.92e10 - 5.81e10i)T^{2} \) |
| 71 | \( 1 + (6.54e4 - 7.79e4i)T + (-2.22e10 - 1.26e11i)T^{2} \) |
| 73 | \( 1 + (-4.64e3 + 2.63e4i)T + (-1.42e11 - 5.17e10i)T^{2} \) |
| 79 | \( 1 + (6.21e4 + 1.09e4i)T + (2.28e11 + 8.31e10i)T^{2} \) |
| 83 | \( 1 + (-2.21e5 - 3.83e5i)T + (-1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + (4.03e5 - 7.12e4i)T + (4.67e11 - 1.69e11i)T^{2} \) |
| 97 | \( 1 + (2.65e5 + 7.28e5i)T + (-6.38e11 + 5.35e11i)T^{2} \) |
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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
−12.36757486744380684532242753719, −11.75029479029262635529483156874, −11.15848426893563072784209011191, −9.685264524918262105691275845069, −8.128204767831079020115100562930, −6.55476116325908785314822378986, −5.65834322176455658199077493243, −4.64819672241830995103971894583, −1.89261474670940379975317776755, −0.16244438622902643805088180446,
1.26147139615772991827697946087, 4.39258186799829253845130622115, 4.93636853936675668804292033083, 6.67962784212093036795955951512, 7.40850465673978431461231157325, 9.740234695409647751328789734259, 10.49673411962433583582804775256, 11.52083505434615033163320841931, 12.19073305394645787639502959116, 13.57426539778288423944886231025
