| L(s) = 1 | + (627. − 362. i)2-s + (2.62e5 − 4.54e5i)4-s + (−1.35e7 − 7.83e6i)5-s + (−1.44e8 − 2.50e8i)7-s − 3.79e8i·8-s − 1.13e10·10-s + (−2.16e8 + 1.24e8i)11-s + (−1.02e11 + 1.77e11i)13-s + (−1.81e11 − 1.04e11i)14-s + (−1.37e11 − 2.38e11i)16-s + 3.27e12i·17-s + 6.83e12·19-s + (−7.11e12 + 4.10e12i)20-s + (−9.04e10 + 1.56e11i)22-s + (1.99e13 + 1.15e13i)23-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.38 − 0.802i)5-s + (−0.511 − 0.886i)7-s − 0.353i·8-s − 1.13·10-s + (−0.00833 + 0.00481i)11-s + (−0.742 + 1.28i)13-s + (−0.626 − 0.361i)14-s + (−0.125 − 0.216i)16-s + 1.62i·17-s + 1.11·19-s + (−0.694 + 0.401i)20-s + (−0.00340 + 0.00589i)22-s + (0.480 + 0.277i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(1.624213681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.624213681\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-627. + 362. i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.35e7 + 7.83e6i)T + (4.76e13 + 8.25e13i)T^{2} \) |
| 7 | \( 1 + (1.44e8 + 2.50e8i)T + (-3.98e16 + 6.91e16i)T^{2} \) |
| 11 | \( 1 + (2.16e8 - 1.24e8i)T + (3.36e20 - 5.82e20i)T^{2} \) |
| 13 | \( 1 + (1.02e11 - 1.77e11i)T + (-9.50e21 - 1.64e22i)T^{2} \) |
| 17 | \( 1 - 3.27e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 6.83e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + (-1.99e13 - 1.15e13i)T + (8.58e26 + 1.48e27i)T^{2} \) |
| 29 | \( 1 + (-4.37e13 + 2.52e13i)T + (8.84e28 - 1.53e29i)T^{2} \) |
| 31 | \( 1 + (5.34e13 - 9.26e13i)T + (-3.35e29 - 5.81e29i)T^{2} \) |
| 37 | \( 1 + 6.08e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (-1.11e16 - 6.42e15i)T + (9.00e31 + 1.56e32i)T^{2} \) |
| 43 | \( 1 + (1.04e16 + 1.80e16i)T + (-2.33e32 + 4.04e32i)T^{2} \) |
| 47 | \( 1 + (-3.71e16 + 2.14e16i)T + (1.38e33 - 2.39e33i)T^{2} \) |
| 53 | \( 1 + 2.87e17iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (3.86e17 + 2.23e17i)T + (1.30e35 + 2.26e35i)T^{2} \) |
| 61 | \( 1 + (4.61e17 + 8.00e17i)T + (-2.54e35 + 4.40e35i)T^{2} \) |
| 67 | \( 1 + (1.15e18 - 2.00e18i)T + (-1.66e36 - 2.87e36i)T^{2} \) |
| 71 | \( 1 + 4.08e17iT - 1.05e37T^{2} \) |
| 73 | \( 1 + 5.70e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (-7.17e18 - 1.24e19i)T + (-4.48e37 + 7.76e37i)T^{2} \) |
| 83 | \( 1 + (-1.76e19 + 1.02e19i)T + (1.20e38 - 2.08e38i)T^{2} \) |
| 89 | \( 1 + 8.44e18iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (-9.95e18 - 1.72e19i)T + (-2.71e39 + 4.70e39i)T^{2} \) |
| 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
−11.53503450223587212671184733716, −10.31873810310320249395036649959, −9.018274588773491446422095764282, −7.68898253291909094415359363282, −6.73878640452311026581509537986, −5.06407882351276167334396138440, −4.07573601775808937383728618652, −3.47186442200893585268328313761, −1.66924313595713280967053406085, −0.55160253923697043434493843669,
0.48691807091790853921470971637, 2.92434391644689728417965426488, 3.03002961562722262479897040940, 4.64014607197058565584370394348, 5.73232641311810052394710828401, 7.17237940856604936751057915476, 7.69635418541364539982544844930, 9.181304509957458727223055831383, 10.68780101206716381655336704046, 11.87611538724074676828874332291
