| L(s) = 1 | + (2.46 + 7.59i)2-s + (0.720 + 0.523i)3-s + (−25.6 + 18.6i)4-s + (−2.24 + 6.89i)5-s + (−2.19 + 6.75i)6-s + (136. − 99.2i)7-s + (1.52 + 1.11i)8-s + (−74.8 − 230. i)9-s − 57.9·10-s + (−328. + 230. i)11-s − 28.2·12-s + (−82.1 − 252. i)13-s + (1.09e3 + 792. i)14-s + (−5.22 + 3.79i)15-s + (−318. + 981. i)16-s + (−212. + 655. i)17-s + ⋯ |
| L(s) = 1 | + (0.436 + 1.34i)2-s + (0.0461 + 0.0335i)3-s + (−0.803 + 0.583i)4-s + (−0.0400 + 0.123i)5-s + (−0.0249 + 0.0766i)6-s + (1.05 − 0.765i)7-s + (0.00844 + 0.00613i)8-s + (−0.308 − 0.947i)9-s − 0.183·10-s + (−0.819 + 0.573i)11-s − 0.0566·12-s + (−0.134 − 0.414i)13-s + (1.48 + 1.08i)14-s + (−0.00599 + 0.00435i)15-s + (−0.311 + 0.958i)16-s + (−0.178 + 0.550i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0632 - 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0632 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.09638 + 1.02906i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09638 + 1.02906i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (328. - 230. i)T \) |
| good | 2 | \( 1 + (-2.46 - 7.59i)T + (-25.8 + 18.8i)T^{2} \) |
| 3 | \( 1 + (-0.720 - 0.523i)T + (75.0 + 231. i)T^{2} \) |
| 5 | \( 1 + (2.24 - 6.89i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-136. + 99.2i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (82.1 + 252. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (212. - 655. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (2.18e3 + 1.59e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 1.40e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-2.16e3 + 1.57e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.33e3 - 4.10e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-636. + 462. i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.04e4 - 7.57e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 2.00e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.52e4 + 1.10e4i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-9.20e3 - 2.83e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.66e4 + 1.20e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-9.60e3 + 2.95e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + 764.T + 1.35e9T^{2} \) |
| 71 | \( 1 + (1.61e4 - 4.96e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-5.72e4 + 4.16e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.16e3 + 3.59e3i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (1.81e3 - 5.57e3i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 - 6.23e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-3.08e4 - 9.49e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−20.04142069775035826826681523077, −17.82358951873353870035490289993, −17.09101596717749274103962898343, −15.28984568344074750750331984059, −14.65006609308429884698902796306, −13.08326891101557017789365653702, −10.83544494899572964117841583847, −8.302161823228420933427431354477, −6.80976888985165892397576486683, −4.77571059272291122672410623345,
2.29422868461102334298940239698, 4.92791763780627249097703593095, 8.344366541620955239873532875744, 10.59685748090480860954576400221, 11.65388498930557931520123944408, 13.08417117995934397681408135709, 14.47704324717733252069084037843, 16.44828427290978620815014946470, 18.41096759211552116629098358268, 19.36431425439548746589689692531
