L(s) = 1 | + (−16.8 + 27.2i)2-s + (183. + 183. i)3-s + (−458. − 915. i)4-s + (3.49e3 + 3.49e3i)5-s + (−8.09e3 + 1.91e3i)6-s + 2.61e4·7-s + (3.26e4 + 2.90e3i)8-s + 8.51e3i·9-s + (−1.53e5 + 3.64e4i)10-s + (−2.36e4 + 2.36e4i)11-s + (8.39e4 − 2.52e5i)12-s + (−1.98e5 + 1.98e5i)13-s + (−4.39e5 + 7.11e5i)14-s + 1.28e6i·15-s + (−6.27e5 + 8.39e5i)16-s − 3.71e4·17-s + ⋯ |
L(s) = 1 | + (−0.525 + 0.850i)2-s + (0.756 + 0.756i)3-s + (−0.447 − 0.894i)4-s + (1.11 + 1.11i)5-s + (−1.04 + 0.246i)6-s + 1.55·7-s + (0.996 + 0.0885i)8-s + 0.144i·9-s + (−1.53 + 0.364i)10-s + (−0.146 + 0.146i)11-s + (0.337 − 1.01i)12-s + (−0.534 + 0.534i)13-s + (−0.816 + 1.32i)14-s + 1.69i·15-s + (−0.598 + 0.801i)16-s − 0.0261·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 - 0.859i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.05761 + 1.85907i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05761 + 1.85907i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (16.8 - 27.2i)T \) |
good | 3 | \( 1 + (-183. - 183. i)T + 5.90e4iT^{2} \) |
| 5 | \( 1 + (-3.49e3 - 3.49e3i)T + 9.76e6iT^{2} \) |
| 7 | \( 1 - 2.61e4T + 2.82e8T^{2} \) |
| 11 | \( 1 + (2.36e4 - 2.36e4i)T - 2.59e10iT^{2} \) |
| 13 | \( 1 + (1.98e5 - 1.98e5i)T - 1.37e11iT^{2} \) |
| 17 | \( 1 + 3.71e4T + 2.01e12T^{2} \) |
| 19 | \( 1 + (2.46e6 + 2.46e6i)T + 6.13e12iT^{2} \) |
| 23 | \( 1 + 9.28e5T + 4.14e13T^{2} \) |
| 29 | \( 1 + (-2.31e7 + 2.31e7i)T - 4.20e14iT^{2} \) |
| 31 | \( 1 - 4.20e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + (7.15e7 + 7.15e7i)T + 4.80e15iT^{2} \) |
| 41 | \( 1 - 3.19e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 + (7.38e7 - 7.38e7i)T - 2.16e16iT^{2} \) |
| 47 | \( 1 - 4.97e6iT - 5.25e16T^{2} \) |
| 53 | \( 1 + (4.22e8 + 4.22e8i)T + 1.74e17iT^{2} \) |
| 59 | \( 1 + (2.04e8 - 2.04e8i)T - 5.11e17iT^{2} \) |
| 61 | \( 1 + (-9.65e8 + 9.65e8i)T - 7.13e17iT^{2} \) |
| 67 | \( 1 + (4.22e7 + 4.22e7i)T + 1.82e18iT^{2} \) |
| 71 | \( 1 - 1.24e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + 1.86e8iT - 4.29e18T^{2} \) |
| 79 | \( 1 - 3.24e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + (-1.80e9 - 1.80e9i)T + 1.55e19iT^{2} \) |
| 89 | \( 1 + 1.74e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 - 1.21e10T + 7.37e19T^{2} \) |
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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
−17.44053742212137538152538124209, −15.46003757666786692346638473728, −14.48901924085005822278099086383, −14.04541948311443095062497209072, −10.86050636871714132372264186520, −9.772888344551055759597523723253, −8.457643737740486601447334389631, −6.71746612117539948432229029962, −4.79820858569671817296510684776, −2.12226380487328081279164261994,
1.30484427699679663775312332397, 2.16671548343905612429176626793, 4.93561849002348737962882239025, 7.964150610018473970286842349694, 8.717820304632127044276220298478, 10.39578277400319738706244242543, 12.24120614143066184466532363909, 13.30201023695092188641268577675, 14.31210448343476836982633654197, 16.89979533019532162893791072128