L(s) = 1 | + (−2.89 − 0.795i)3-s + (−0.355 + 0.615i)5-s + (2.70 − 1.56i)7-s + (7.73 + 4.60i)9-s + (−14.3 + 8.30i)11-s + (9.17 − 15.8i)13-s + (1.51 − 1.49i)15-s − 9.69·17-s + 8.20i·19-s + (−9.06 + 2.36i)21-s + (1.94 + 1.12i)23-s + (12.2 + 21.2i)25-s + (−18.7 − 19.4i)27-s + (20.8 + 36.0i)29-s + (−21.6 − 12.4i)31-s + ⋯ |
L(s) = 1 | + (−0.964 − 0.265i)3-s + (−0.0710 + 0.123i)5-s + (0.386 − 0.223i)7-s + (0.859 + 0.511i)9-s + (−1.30 + 0.754i)11-s + (0.705 − 1.22i)13-s + (0.101 − 0.0998i)15-s − 0.570·17-s + 0.431i·19-s + (−0.431 + 0.112i)21-s + (0.0847 + 0.0489i)23-s + (0.489 + 0.848i)25-s + (−0.692 − 0.721i)27-s + (0.717 + 1.24i)29-s + (−0.697 − 0.402i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.167179593\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167179593\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.89 + 0.795i)T \) |
good | 5 | \( 1 + (0.355 - 0.615i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-2.70 + 1.56i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (14.3 - 8.30i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-9.17 + 15.8i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 9.69T + 289T^{2} \) |
| 19 | \( 1 - 8.20iT - 361T^{2} \) |
| 23 | \( 1 + (-1.94 - 1.12i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-20.8 - 36.0i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (21.6 + 12.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 40.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-25.6 + 44.5i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-56.6 + 32.7i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-29.2 + 16.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 90.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-66.2 - 38.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.35 + 2.35i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-34.5 - 19.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 38.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-94.4 + 54.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (113. - 65.5i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 38.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + (12.1 + 21.1i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.74425597016962521893753827887, −10.02487965935024695825022403098, −8.658426105840680093241306753861, −7.61029326225972205615732844238, −7.10086678826113785964657925522, −5.73485135442226391156130017109, −5.21108521205783781921209527291, −4.02992766972515496114690746272, −2.43606049439942400061558970910, −0.890892422912187226533729607025,
0.72226661195392696119481537695, 2.45194575907924953574273896882, 4.10709722916793157082006918662, 4.89066166515532263133420081305, 5.91560715456244566475820897204, 6.64571945070352729922136439564, 7.86668181190866156083359275585, 8.766403210345558621238812512203, 9.709357922680699539578116746237, 10.81654087924467820374533665949