| L(s) = 1 | + (−14.4 + 10.5i)2-s + (−4.17 − 12.8i)3-s + (−59.1 + 182. i)4-s + (−591. − 429. i)5-s + (195. + 142. i)6-s + (2.10e3 − 6.47e3i)7-s + (−3.89e3 − 1.19e4i)8-s + (1.57e4 − 1.14e4i)9-s + 1.30e4·10-s + (−3.25e4 − 3.60e4i)11-s + 2.58e3·12-s + (−6.74e4 + 4.90e4i)13-s + (3.76e4 + 1.15e5i)14-s + (−3.05e3 + 9.40e3i)15-s + (1.03e5 + 7.49e4i)16-s + (1.28e4 + 9.36e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.640 + 0.465i)2-s + (−0.0297 − 0.0916i)3-s + (−0.115 + 0.355i)4-s + (−0.423 − 0.307i)5-s + (0.0617 + 0.0448i)6-s + (0.331 − 1.01i)7-s + (−0.335 − 1.03i)8-s + (0.801 − 0.582i)9-s + 0.413·10-s + (−0.669 − 0.742i)11-s + 0.0360·12-s + (−0.655 + 0.475i)13-s + (0.261 + 0.806i)14-s + (−0.0155 + 0.0479i)15-s + (0.393 + 0.285i)16-s + (0.0374 + 0.0271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0813 + 0.996i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0813 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.466806 - 0.430243i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.466806 - 0.430243i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (3.25e4 + 3.60e4i)T \) |
| good | 2 | \( 1 + (14.4 - 10.5i)T + (158. - 486. i)T^{2} \) |
| 3 | \( 1 + (4.17 + 12.8i)T + (-1.59e4 + 1.15e4i)T^{2} \) |
| 5 | \( 1 + (591. + 429. i)T + (6.03e5 + 1.85e6i)T^{2} \) |
| 7 | \( 1 + (-2.10e3 + 6.47e3i)T + (-3.26e7 - 2.37e7i)T^{2} \) |
| 13 | \( 1 + (6.74e4 - 4.90e4i)T + (3.27e9 - 1.00e10i)T^{2} \) |
| 17 | \( 1 + (-1.28e4 - 9.36e3i)T + (3.66e10 + 1.12e11i)T^{2} \) |
| 19 | \( 1 + (4.42e4 + 1.36e5i)T + (-2.61e11 + 1.89e11i)T^{2} \) |
| 23 | \( 1 + 1.08e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + (-1.08e6 + 3.34e6i)T + (-1.17e13 - 8.52e12i)T^{2} \) |
| 31 | \( 1 + (1.91e6 - 1.39e6i)T + (8.17e12 - 2.51e13i)T^{2} \) |
| 37 | \( 1 + (6.21e6 - 1.91e7i)T + (-1.05e14 - 7.63e13i)T^{2} \) |
| 41 | \( 1 + (7.52e6 + 2.31e7i)T + (-2.64e14 + 1.92e14i)T^{2} \) |
| 43 | \( 1 - 1.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (1.55e7 + 4.77e7i)T + (-9.05e14 + 6.57e14i)T^{2} \) |
| 53 | \( 1 + (6.34e6 - 4.60e6i)T + (1.01e15 - 3.13e15i)T^{2} \) |
| 59 | \( 1 + (1.25e7 - 3.86e7i)T + (-7.00e15 - 5.09e15i)T^{2} \) |
| 61 | \( 1 + (-1.49e8 - 1.08e8i)T + (3.61e15 + 1.11e16i)T^{2} \) |
| 67 | \( 1 + 5.86e6T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-2.05e8 - 1.48e8i)T + (1.41e16 + 4.36e16i)T^{2} \) |
| 73 | \( 1 + (3.17e7 - 9.76e7i)T + (-4.76e16 - 3.46e16i)T^{2} \) |
| 79 | \( 1 + (-3.83e8 + 2.78e8i)T + (3.70e16 - 1.13e17i)T^{2} \) |
| 83 | \( 1 + (-9.07e7 - 6.59e7i)T + (5.77e16 + 1.77e17i)T^{2} \) |
| 89 | \( 1 + 4.55e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (1.26e9 - 9.22e8i)T + (2.34e17 - 7.23e17i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.84658825470420813138891088490, −16.70296986303629488440101485555, −15.61885467409511104790708683584, −13.60184717028484323904425902969, −12.11159916522832210331024745987, −10.05896988248989046195046370647, −8.276890405337749120553665081662, −7.03156838203867986170635769831, −4.07858607824914720659668820966, −0.46495336666670154341221284074,
2.10123526536415876305217438111, 5.18022228083612170023901818031, 7.83647326635414141195195979591, 9.638588205834173245398778638947, 10.90336005635773196217436059720, 12.49469378085141543431410051899, 14.64287853995575153561033545484, 15.72744417022195790771654546092, 17.81437532125844145729557845003, 18.66846920946495201210627468057
