| L(s) = 1 | + (24.5 + 17.8i)2-s + (−45.7 + 140. i)3-s + (127. + 391. i)4-s + (−601. + 437. i)5-s + (−3.64e3 + 2.64e3i)6-s + (430. + 1.32e3i)7-s + (937. − 2.88e3i)8-s + (−1.82e3 − 1.32e3i)9-s − 2.26e4·10-s + (2.76e3 + 4.84e4i)11-s − 6.10e4·12-s + (7.21e4 + 5.24e4i)13-s + (−1.30e4 + 4.02e4i)14-s + (−3.40e4 − 1.04e5i)15-s + (2.45e5 − 1.78e5i)16-s + (2.37e5 − 1.72e5i)17-s + ⋯ |
| L(s) = 1 | + (1.08 + 0.789i)2-s + (−0.326 + 1.00i)3-s + (0.248 + 0.765i)4-s + (−0.430 + 0.312i)5-s + (−1.14 + 0.833i)6-s + (0.0677 + 0.208i)7-s + (0.0809 − 0.249i)8-s + (−0.0925 − 0.0672i)9-s − 0.715·10-s + (0.0569 + 0.998i)11-s − 0.849·12-s + (0.701 + 0.509i)13-s + (−0.0910 + 0.280i)14-s + (−0.173 − 0.534i)15-s + (0.936 − 0.680i)16-s + (0.690 − 0.501i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.965694 + 2.10094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.965694 + 2.10094i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-2.76e3 - 4.84e4i)T \) |
| good | 2 | \( 1 + (-24.5 - 17.8i)T + (158. + 486. i)T^{2} \) |
| 3 | \( 1 + (45.7 - 140. i)T + (-1.59e4 - 1.15e4i)T^{2} \) |
| 5 | \( 1 + (601. - 437. i)T + (6.03e5 - 1.85e6i)T^{2} \) |
| 7 | \( 1 + (-430. - 1.32e3i)T + (-3.26e7 + 2.37e7i)T^{2} \) |
| 13 | \( 1 + (-7.21e4 - 5.24e4i)T + (3.27e9 + 1.00e10i)T^{2} \) |
| 17 | \( 1 + (-2.37e5 + 1.72e5i)T + (3.66e10 - 1.12e11i)T^{2} \) |
| 19 | \( 1 + (-1.89e5 + 5.82e5i)T + (-2.61e11 - 1.89e11i)T^{2} \) |
| 23 | \( 1 - 3.85e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + (1.33e6 + 4.10e6i)T + (-1.17e13 + 8.52e12i)T^{2} \) |
| 31 | \( 1 + (4.15e6 + 3.01e6i)T + (8.17e12 + 2.51e13i)T^{2} \) |
| 37 | \( 1 + (-5.92e6 - 1.82e7i)T + (-1.05e14 + 7.63e13i)T^{2} \) |
| 41 | \( 1 + (-7.78e6 + 2.39e7i)T + (-2.64e14 - 1.92e14i)T^{2} \) |
| 43 | \( 1 + 2.90e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-8.55e6 + 2.63e7i)T + (-9.05e14 - 6.57e14i)T^{2} \) |
| 53 | \( 1 + (2.56e7 + 1.86e7i)T + (1.01e15 + 3.13e15i)T^{2} \) |
| 59 | \( 1 + (-3.08e7 - 9.50e7i)T + (-7.00e15 + 5.09e15i)T^{2} \) |
| 61 | \( 1 + (-1.15e8 + 8.37e7i)T + (3.61e15 - 1.11e16i)T^{2} \) |
| 67 | \( 1 - 9.99e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + (2.15e8 - 1.56e8i)T + (1.41e16 - 4.36e16i)T^{2} \) |
| 73 | \( 1 + (-3.82e7 - 1.17e8i)T + (-4.76e16 + 3.46e16i)T^{2} \) |
| 79 | \( 1 + (-2.34e8 - 1.70e8i)T + (3.70e16 + 1.13e17i)T^{2} \) |
| 83 | \( 1 + (-3.24e8 + 2.36e8i)T + (5.77e16 - 1.77e17i)T^{2} \) |
| 89 | \( 1 + 7.83e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (2.31e8 + 1.68e8i)T + (2.34e17 + 7.23e17i)T^{2} \) |
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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
−18.78182017571645000385974512718, −16.80761397793965169345290026897, −15.60471072078790053340669847179, −14.98785138704888848428479750835, −13.37416225982331540249154747939, −11.55080124719829837425019072716, −9.765973129139572551250704880894, −7.15180678660246246889042559832, −5.23929842827147776960144031286, −3.93311747308671758024253141226,
1.22213059233694565986227611701, 3.61997539049426198359135613655, 5.80550279685105643765522183235, 8.012535617734525229505441948644, 10.93919511348661436595324769325, 12.22598827099745075987918845586, 13.05879406977573330721315742589, 14.35546132124014897659190052016, 16.42136945582985418120701007860, 18.14191274474203389163888997941
