| L(s) = 1 | + (1.38e3 + 2.40e3i)2-s + (5.08e4 − 8.87e4i)3-s + (−2.80e6 + 4.85e6i)4-s + (1.57e7 − 2.73e7i)5-s + (2.83e8 − 1.01e6i)6-s + (−2.72e8 − 4.71e8i)7-s − 9.75e9·8-s + (−5.29e9 − 9.02e9i)9-s + 8.75e10·10-s + (−4.94e10 − 8.55e10i)11-s + (2.88e11 + 4.95e11i)12-s + (2.11e11 − 3.65e11i)13-s + (7.56e11 − 1.31e12i)14-s + (−1.62e12 − 2.78e12i)15-s + (−7.65e12 − 1.32e13i)16-s + 8.22e11·17-s + ⋯ |
| L(s) = 1 | + (0.958 + 1.66i)2-s + (0.496 − 0.867i)3-s + (−1.33 + 2.31i)4-s + (0.721 − 1.25i)5-s + (1.91 − 0.00682i)6-s + (−0.364 − 0.631i)7-s − 3.21·8-s + (−0.506 − 0.862i)9-s + 2.76·10-s + (−0.574 − 0.994i)11-s + (1.34 + 2.31i)12-s + (0.424 − 0.735i)13-s + (0.699 − 1.21i)14-s + (−0.726 − 1.24i)15-s + (−1.74 − 3.01i)16-s + 0.0989·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(3.007864296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.007864296\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.08e4 + 8.87e4i)T \) |
| good | 2 | \( 1 + (-1.38e3 - 2.40e3i)T + (-1.04e6 + 1.81e6i)T^{2} \) |
| 5 | \( 1 + (-1.57e7 + 2.73e7i)T + (-2.38e14 - 4.12e14i)T^{2} \) |
| 7 | \( 1 + (2.72e8 + 4.71e8i)T + (-2.79e17 + 4.83e17i)T^{2} \) |
| 11 | \( 1 + (4.94e10 + 8.55e10i)T + (-3.70e21 + 6.40e21i)T^{2} \) |
| 13 | \( 1 + (-2.11e11 + 3.65e11i)T + (-1.23e23 - 2.13e23i)T^{2} \) |
| 17 | \( 1 - 8.22e11T + 6.90e25T^{2} \) |
| 19 | \( 1 + 2.94e10T + 7.14e26T^{2} \) |
| 23 | \( 1 + (-5.19e12 + 8.99e12i)T + (-1.97e28 - 3.41e28i)T^{2} \) |
| 29 | \( 1 + (-1.47e15 - 2.56e15i)T + (-2.56e30 + 4.44e30i)T^{2} \) |
| 31 | \( 1 + (2.19e15 - 3.79e15i)T + (-1.04e31 - 1.80e31i)T^{2} \) |
| 37 | \( 1 - 4.72e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + (1.43e15 - 2.48e15i)T + (-3.69e33 - 6.39e33i)T^{2} \) |
| 43 | \( 1 + (4.07e16 + 7.06e16i)T + (-1.00e34 + 1.73e34i)T^{2} \) |
| 47 | \( 1 + (-8.34e16 - 1.44e17i)T + (-6.50e34 + 1.12e35i)T^{2} \) |
| 53 | \( 1 + 1.19e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + (-2.85e18 + 4.94e18i)T + (-7.70e36 - 1.33e37i)T^{2} \) |
| 61 | \( 1 + (8.78e17 + 1.52e18i)T + (-1.55e37 + 2.68e37i)T^{2} \) |
| 67 | \( 1 + (-1.15e19 + 1.99e19i)T + (-1.11e38 - 1.92e38i)T^{2} \) |
| 71 | \( 1 + 2.48e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.60e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + (-8.64e18 - 1.49e19i)T + (-3.54e39 + 6.13e39i)T^{2} \) |
| 83 | \( 1 + (5.57e19 + 9.64e19i)T + (-9.99e39 + 1.73e40i)T^{2} \) |
| 89 | \( 1 - 2.67e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + (-3.80e20 - 6.59e20i)T + (-2.63e41 + 4.56e41i)T^{2} \) |
| 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
−15.97689789357083378255082536193, −14.20820755739889030481406294973, −13.26733037032343322014196807134, −12.70552742023694883694979930942, −8.887872719080293887995989781964, −7.901069131344741061000606024509, −6.34470764786327803445929959099, −5.21530958524894914690879092521, −3.33766535611560867322789205853, −0.67310914545094423095564701573,
2.16185796259920626072451453472, 2.82230195777252524702748189161, 4.28686760458887673367973904424, 5.88980995067774904749706098805, 9.479979137829027789930389881176, 10.23831823108289334246680667448, 11.44701473071343808899347734527, 13.22371237621014478736159520078, 14.36796855966820797145987678869, 15.26793033878925872106236550463
