| L(s) = 1 | + (42.4 − 15.4i)3-s + (−0.750 + 4.25i)5-s + (0.108 + 0.188i)7-s + (−108. + 91.1i)9-s + (3.43e3 − 5.94e3i)11-s + (1.01e4 + 3.68e3i)13-s + (33.9 + 192. i)15-s + (−1.55e4 − 1.30e4i)17-s + (2.95e4 − 4.48e3i)19-s + (7.53 + 6.32i)21-s + (−1.18e4 − 6.71e4i)23-s + (7.33e4 + 2.67e4i)25-s + (−5.26e4 + 9.12e4i)27-s + (1.31e3 − 1.10e3i)29-s + (−8.65e4 − 1.49e5i)31-s + ⋯ |
| L(s) = 1 | + (0.908 − 0.330i)3-s + (−0.00268 + 0.0152i)5-s + (0.000119 + 0.000207i)7-s + (−0.0496 + 0.0416i)9-s + (0.778 − 1.34i)11-s + (1.27 + 0.465i)13-s + (0.00259 + 0.0147i)15-s + (−0.769 − 0.645i)17-s + (0.988 − 0.149i)19-s + (0.000177 + 0.000149i)21-s + (−0.202 − 1.15i)23-s + (0.939 + 0.341i)25-s + (−0.514 + 0.891i)27-s + (0.0100 − 0.00841i)29-s + (−0.522 − 0.904i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.48772 - 1.17358i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48772 - 1.17358i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.95e4 + 4.48e3i)T \) |
| good | 3 | \( 1 + (-42.4 + 15.4i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (0.750 - 4.25i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-0.108 - 0.188i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-3.43e3 + 5.94e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-1.01e4 - 3.68e3i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (1.55e4 + 1.30e4i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (1.18e4 + 6.71e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (-1.31e3 + 1.10e3i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (8.65e4 + 1.49e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 3.39e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (7.76e4 - 2.82e4i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-1.10e5 + 6.28e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-9.82e5 + 8.24e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (-2.48e5 - 1.40e6i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (1.09e6 + 9.21e5i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-3.01e5 - 1.71e6i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (3.14e6 - 2.63e6i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (4.26e5 - 2.41e6i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (4.61e6 - 1.67e6i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (-2.50e5 + 9.11e4i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-2.05e6 - 3.56e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (3.89e6 + 1.41e6i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (-1.02e7 - 8.63e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−13.39923060527447458100834471223, −11.71231336780502546784640365143, −10.85813119722075671673103384097, −9.005293290565214732915234364806, −8.625238673418579358193489036665, −7.14903359358813296111056859704, −5.83050863464116202851219492214, −3.89158724392497823708830530234, −2.63349442919252056836058258753, −0.953920711191462176760862312573,
1.47627875144045275988887490808, 3.16249291323483339476664877529, 4.32557670535198091260063592222, 6.12774988277625577174785545466, 7.58941508979215262971355774537, 8.830824713092152407184336311363, 9.589551062789754010369537312358, 10.92649340290906479700157096838, 12.21234699214189239097597479606, 13.37422330102427083881969930563
