| L(s) = 1 | + (−5.91 − 18.2i)2-s + (−54.9 − 39.8i)3-s + (1.36e3 − 988. i)4-s + (−614. + 1.89e3i)5-s + (−401. + 1.23e3i)6-s + (1.99e4 − 1.44e4i)7-s + (−5.77e4 − 4.19e4i)8-s + (−5.33e4 − 1.64e5i)9-s + 3.81e4·10-s + (−5.32e5 + 3.69e4i)11-s − 1.14e5·12-s + (−6.50e5 − 2.00e6i)13-s + (−3.81e5 − 2.76e5i)14-s + (1.09e5 − 7.93e4i)15-s + (6.41e5 − 1.97e6i)16-s + (−1.06e5 + 3.26e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.130 − 0.402i)2-s + (−0.130 − 0.0947i)3-s + (0.664 − 0.482i)4-s + (−0.0880 + 0.270i)5-s + (−0.0210 + 0.0648i)6-s + (0.447 − 0.325i)7-s + (−0.623 − 0.452i)8-s + (−0.300 − 0.926i)9-s + 0.120·10-s + (−0.997 + 0.0691i)11-s − 0.132·12-s + (−0.485 − 1.49i)13-s + (−0.189 − 0.137i)14-s + (0.0371 − 0.0269i)15-s + (0.152 − 0.470i)16-s + (−0.0181 + 0.0558i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.660863 - 1.26023i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.660863 - 1.26023i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (5.32e5 - 3.69e4i)T \) |
| good | 2 | \( 1 + (5.91 + 18.2i)T + (-1.65e3 + 1.20e3i)T^{2} \) |
| 3 | \( 1 + (54.9 + 39.8i)T + (5.47e4 + 1.68e5i)T^{2} \) |
| 5 | \( 1 + (614. - 1.89e3i)T + (-3.95e7 - 2.87e7i)T^{2} \) |
| 7 | \( 1 + (-1.99e4 + 1.44e4i)T + (6.11e8 - 1.88e9i)T^{2} \) |
| 13 | \( 1 + (6.50e5 + 2.00e6i)T + (-1.44e12 + 1.05e12i)T^{2} \) |
| 17 | \( 1 + (1.06e5 - 3.26e5i)T + (-2.77e13 - 2.01e13i)T^{2} \) |
| 19 | \( 1 + (-9.77e6 - 7.10e6i)T + (3.59e13 + 1.10e14i)T^{2} \) |
| 23 | \( 1 + 2.09e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + (-7.43e7 + 5.40e7i)T + (3.77e15 - 1.16e16i)T^{2} \) |
| 31 | \( 1 + (6.61e7 + 2.03e8i)T + (-2.05e16 + 1.49e16i)T^{2} \) |
| 37 | \( 1 + (1.52e8 - 1.10e8i)T + (5.49e16 - 1.69e17i)T^{2} \) |
| 41 | \( 1 + (-6.23e8 - 4.53e8i)T + (1.70e17 + 5.23e17i)T^{2} \) |
| 43 | \( 1 - 6.61e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.41e9 - 1.03e9i)T + (7.63e17 + 2.35e18i)T^{2} \) |
| 53 | \( 1 + (1.77e8 + 5.44e8i)T + (-7.49e18 + 5.44e18i)T^{2} \) |
| 59 | \( 1 + (-2.35e9 + 1.70e9i)T + (9.31e18 - 2.86e19i)T^{2} \) |
| 61 | \( 1 + (-4.77e8 + 1.47e9i)T + (-3.52e19 - 2.55e19i)T^{2} \) |
| 67 | \( 1 - 9.33e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (8.69e9 - 2.67e10i)T + (-1.86e20 - 1.35e20i)T^{2} \) |
| 73 | \( 1 + (-1.00e10 + 7.31e9i)T + (9.69e19 - 2.98e20i)T^{2} \) |
| 79 | \( 1 + (5.52e9 + 1.70e10i)T + (-6.05e20 + 4.39e20i)T^{2} \) |
| 83 | \( 1 + (-1.49e10 + 4.60e10i)T + (-1.04e21 - 7.56e20i)T^{2} \) |
| 89 | \( 1 + 5.63e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (2.76e10 + 8.51e10i)T + (-5.78e21 + 4.20e21i)T^{2} \) |
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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
−17.56887667167352644098767613213, −15.67129982251189271316712150846, −14.63539838099632532736603407843, −12.57608160989915923461243862576, −11.15382689958407003468900129834, −9.951633590960949069255588166409, −7.58993430626981948711450559048, −5.71099778077666894975454652890, −2.90853027306182064307726091108, −0.74586482165188647238384859344,
2.39110806800647674995966761883, 5.11822850402029576641804245896, 7.20949162105283982088765229311, 8.644529423513183090626356709179, 10.94542106713947046343350088860, 12.21433642196039937999909419557, 14.10220392091277707733612642782, 15.82255849747034774068053140318, 16.60365471992234707215712706933, 18.02113337931829247158136794764
