| L(s) = 1 | + (2.40 − 7.41i)2-s + (23.7 − 17.2i)3-s + (54.3 + 39.4i)4-s + (−139. − 429. i)5-s + (−70.7 − 217. i)6-s + (40.1 + 29.1i)7-s + (1.23e3 − 894. i)8-s + (−409. + 1.26e3i)9-s − 3.52e3·10-s + (3.96e3 + 1.93e3i)11-s + 1.97e3·12-s + (−3.56e3 + 1.09e4i)13-s + (313. − 227. i)14-s + (−1.07e4 − 7.79e3i)15-s + (−1.01e3 − 3.11e3i)16-s + (366. + 1.12e3i)17-s + ⋯ |
| L(s) = 1 | + (0.213 − 0.655i)2-s + (0.507 − 0.368i)3-s + (0.424 + 0.308i)4-s + (−0.499 − 1.53i)5-s + (−0.133 − 0.411i)6-s + (0.0442 + 0.0321i)7-s + (0.850 − 0.617i)8-s + (−0.187 + 0.576i)9-s − 1.11·10-s + (0.899 + 0.437i)11-s + 0.329·12-s + (−0.450 + 1.38i)13-s + (0.0305 − 0.0221i)14-s + (−0.820 − 0.596i)15-s + (−0.0617 − 0.189i)16-s + (0.0181 + 0.0557i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.48527 - 1.18888i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48527 - 1.18888i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-3.96e3 - 1.93e3i)T \) |
| good | 2 | \( 1 + (-2.40 + 7.41i)T + (-103. - 75.2i)T^{2} \) |
| 3 | \( 1 + (-23.7 + 17.2i)T + (675. - 2.07e3i)T^{2} \) |
| 5 | \( 1 + (139. + 429. i)T + (-6.32e4 + 4.59e4i)T^{2} \) |
| 7 | \( 1 + (-40.1 - 29.1i)T + (2.54e5 + 7.83e5i)T^{2} \) |
| 13 | \( 1 + (3.56e3 - 1.09e4i)T + (-5.07e7 - 3.68e7i)T^{2} \) |
| 17 | \( 1 + (-366. - 1.12e3i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-2.16e4 + 1.57e4i)T + (2.76e8 - 8.50e8i)T^{2} \) |
| 23 | \( 1 + 5.77e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (6.66e4 + 4.84e4i)T + (5.33e9 + 1.64e10i)T^{2} \) |
| 31 | \( 1 + (-5.05e3 + 1.55e4i)T + (-2.22e10 - 1.61e10i)T^{2} \) |
| 37 | \( 1 + (2.02e5 + 1.47e5i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-6.10e5 + 4.43e5i)T + (6.01e10 - 1.85e11i)T^{2} \) |
| 43 | \( 1 - 2.74e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + (5.04e5 - 3.66e5i)T + (1.56e11 - 4.81e11i)T^{2} \) |
| 53 | \( 1 + (6.35e5 - 1.95e6i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (6.64e5 + 4.83e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (-2.36e5 - 7.27e5i)T + (-2.54e12 + 1.84e12i)T^{2} \) |
| 67 | \( 1 + 2.88e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (1.06e6 + 3.27e6i)T + (-7.35e12 + 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-1.27e5 - 9.28e4i)T + (3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (2.44e6 - 7.53e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (2.33e5 + 7.18e5i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 - 7.30e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-1.72e6 + 5.31e6i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−19.39937312015169728351031597780, −16.97418660390518759588709812365, −16.07199614341466411779494237402, −13.89697916715302102207564522278, −12.51511582941819512898796560637, −11.60859807651279129323684537941, −9.144963591589437606171008130082, −7.53462655235793948477134923270, −4.33069608020485367950618605997, −1.75339770711246910208316507109,
3.25217624736074457808057685514, 6.22838794032736005663629121506, 7.72816573148964331758426163280, 10.15545208334706133875790167056, 11.56110360300565991419284619730, 14.32459657801996281212790157160, 14.81623054912972321484985205966, 15.93380262569170313415240217324, 17.81549985206250072842869789279, 19.39095371720639956200300572428
