| L(s) = 1 | + (−3.29 + 10.1i)2-s + (2.26 − 1.64i)3-s + (−65.9 − 47.9i)4-s + (20.4 + 62.9i)5-s + (9.21 + 28.3i)6-s + (58.3 + 42.3i)7-s + (427. − 310. i)8-s + (−72.6 + 223. i)9-s − 704.·10-s + (369. − 156. i)11-s − 228.·12-s + (177. − 545. i)13-s + (−621. + 451. i)14-s + (149. + 108. i)15-s + (931. + 2.86e3i)16-s + (−84.6 − 260. i)17-s + ⋯ |
| L(s) = 1 | + (−0.582 + 1.79i)2-s + (0.145 − 0.105i)3-s + (−2.06 − 1.49i)4-s + (0.365 + 1.12i)5-s + (0.104 + 0.321i)6-s + (0.449 + 0.326i)7-s + (2.35 − 1.71i)8-s + (−0.299 + 0.920i)9-s − 2.22·10-s + (0.920 − 0.389i)11-s − 0.457·12-s + (0.290 − 0.895i)13-s + (−0.847 + 0.615i)14-s + (0.171 + 0.124i)15-s + (0.910 + 2.80i)16-s + (−0.0710 − 0.218i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.225408 + 0.889778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.225408 + 0.889778i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-369. + 156. i)T \) |
| good | 2 | \( 1 + (3.29 - 10.1i)T + (-25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (-2.26 + 1.64i)T + (75.0 - 231. i)T^{2} \) |
| 5 | \( 1 + (-20.4 - 62.9i)T + (-2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-58.3 - 42.3i)T + (5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-177. + 545. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (84.6 + 260. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-404. + 293. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + 605.T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-43.2 - 31.4i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (1.25e3 - 3.85e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (1.15e4 + 8.36e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-6.37e3 + 4.63e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.02e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.25e4 + 9.13e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (6.00e3 - 1.84e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-3.53e4 - 2.56e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (8.31e3 + 2.55e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + 3.01e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-5.59e3 - 1.72e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (2.47e4 + 1.80e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (3.37e3 - 1.03e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (3.20e4 + 9.87e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + 9.67e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (1.93e3 - 5.94e3i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−19.40642488578253501963007881393, −18.28541655201488970338235144027, −17.36977315595963192626440202333, −15.92443222316994990122432322405, −14.59014479011876174912718776734, −13.85046622771888350699054508505, −10.59091570522199539969001437358, −8.728758851782435618765554365692, −7.24304930539911892877879046520, −5.67517565940494060602396876471,
1.33125153848991082055055440063, 4.16145927864414146810820003347, 8.723252763011034839520095445159, 9.620996927796509952737483227692, 11.50743636435408358730677539287, 12.55027968032142721723300178211, 14.04882669541394861178216046172, 16.91374531954605161858790095915, 17.77180767739147146953564603672, 19.33862046231780883012143662900
