L(s) = 1 | + (−137. − 137. i)3-s + (−3.89e3 + 3.89e3i)5-s − 3.79e4i·7-s − 1.39e5i·9-s + (6.81e4 − 6.81e4i)11-s + (1.09e5 + 1.09e5i)13-s + 1.06e6·15-s + 8.85e6·17-s + (7.12e5 + 7.12e5i)19-s + (−5.19e6 + 5.19e6i)21-s − 2.69e7i·23-s + 1.84e7i·25-s + (−4.34e7 + 4.34e7i)27-s + (1.10e8 + 1.10e8i)29-s − 1.15e8·31-s + ⋯ |
L(s) = 1 | + (−0.325 − 0.325i)3-s + (−0.558 + 0.558i)5-s − 0.853i·7-s − 0.787i·9-s + (0.127 − 0.127i)11-s + (0.0820 + 0.0820i)13-s + 0.363·15-s + 1.51·17-s + (0.0660 + 0.0660i)19-s + (−0.277 + 0.277i)21-s − 0.873i·23-s + 0.377i·25-s + (−0.582 + 0.582i)27-s + (0.999 + 0.999i)29-s − 0.725·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(1.312986485\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.312986485\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (137. + 137. i)T + 1.77e5iT^{2} \) |
| 5 | \( 1 + (3.89e3 - 3.89e3i)T - 4.88e7iT^{2} \) |
| 7 | \( 1 + 3.79e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (-6.81e4 + 6.81e4i)T - 2.85e11iT^{2} \) |
| 13 | \( 1 + (-1.09e5 - 1.09e5i)T + 1.79e12iT^{2} \) |
| 17 | \( 1 - 8.85e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (-7.12e5 - 7.12e5i)T + 1.16e14iT^{2} \) |
| 23 | \( 1 + 2.69e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (-1.10e8 - 1.10e8i)T + 1.22e16iT^{2} \) |
| 31 | \( 1 + 1.15e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + (3.15e8 - 3.15e8i)T - 1.77e17iT^{2} \) |
| 41 | \( 1 + 8.66e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (-3.13e8 + 3.13e8i)T - 9.29e17iT^{2} \) |
| 47 | \( 1 - 1.93e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + (8.24e8 - 8.24e8i)T - 9.26e18iT^{2} \) |
| 59 | \( 1 + (-7.47e9 + 7.47e9i)T - 3.01e19iT^{2} \) |
| 61 | \( 1 + (-6.44e8 - 6.44e8i)T + 4.35e19iT^{2} \) |
| 67 | \( 1 + (5.39e9 + 5.39e9i)T + 1.22e20iT^{2} \) |
| 71 | \( 1 - 2.90e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 1.12e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 2.03e7T + 7.47e20T^{2} \) |
| 83 | \( 1 + (2.10e10 + 2.10e10i)T + 1.28e21iT^{2} \) |
| 89 | \( 1 + 7.01e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 3.85e10T + 7.15e21T^{2} \) |
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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
−10.89180861928765648307879175793, −10.05973422439517320926062286711, −8.698908734063190770170565632923, −7.39601008810688380103714442389, −6.80371439322589311466996081743, −5.52593623991437491842385706037, −3.96229113235906409177790548646, −3.15821136209258676670972038469, −1.29590396063154078753735455756, −0.37200276052773935253892040212,
1.04252770905877169663648613436, 2.48157555282153582238430389400, 3.92188819242368757349776192391, 5.08801063849975551967129375706, 5.85802175566301471496573281036, 7.54742609633051023905918472499, 8.365834631420284702081703180830, 9.516585653371336343684233723590, 10.54338581756925774755732852187, 11.75376699776924763260178827990