| L(s) = 1 | + (351. − 202. i)2-s + (8.21e3 + 7.85e3i)3-s + (1.68e4 − 2.91e4i)4-s + (−3.02e5 − 5.23e5i)5-s + (4.48e6 + 1.09e6i)6-s + (1.44e7 + 4.98e6i)7-s + 3.95e7i·8-s + (5.72e6 + 1.29e8i)9-s + (−2.12e8 − 1.22e8i)10-s + (−5.40e8 − 3.11e8i)11-s + (3.67e8 − 1.07e8i)12-s + 2.26e9i·13-s + (6.07e9 − 1.17e9i)14-s + (1.62e9 − 6.67e9i)15-s + (1.02e10 + 1.77e10i)16-s + (−1.73e10 + 3.00e10i)17-s + ⋯ |
| L(s) = 1 | + (0.971 − 0.560i)2-s + (0.722 + 0.691i)3-s + (0.128 − 0.222i)4-s + (−0.345 − 0.599i)5-s + (1.08 + 0.266i)6-s + (0.945 + 0.326i)7-s + 0.832i·8-s + (0.0443 + 0.999i)9-s + (−0.671 − 0.387i)10-s + (−0.759 − 0.438i)11-s + (0.246 − 0.0720i)12-s + 0.770i·13-s + (1.10 − 0.212i)14-s + (0.164 − 0.672i)15-s + (0.595 + 1.03i)16-s + (−0.603 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(3.956330807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.956330807\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-8.21e3 - 7.85e3i)T \) |
| 7 | \( 1 + (-1.44e7 - 4.98e6i)T \) |
| good | 2 | \( 1 + (-351. + 202. i)T + (6.55e4 - 1.13e5i)T^{2} \) |
| 5 | \( 1 + (3.02e5 + 5.23e5i)T + (-3.81e11 + 6.60e11i)T^{2} \) |
| 11 | \( 1 + (5.40e8 + 3.11e8i)T + (2.52e17 + 4.37e17i)T^{2} \) |
| 13 | \( 1 - 2.26e9iT - 8.65e18T^{2} \) |
| 17 | \( 1 + (1.73e10 - 3.00e10i)T + (-4.13e20 - 7.16e20i)T^{2} \) |
| 19 | \( 1 + (-1.02e11 + 5.90e10i)T + (2.74e21 - 4.74e21i)T^{2} \) |
| 23 | \( 1 + (2.54e11 - 1.46e11i)T + (7.05e22 - 1.22e23i)T^{2} \) |
| 29 | \( 1 - 4.91e12iT - 7.25e24T^{2} \) |
| 31 | \( 1 + (-2.83e11 - 1.63e11i)T + (1.12e25 + 1.95e25i)T^{2} \) |
| 37 | \( 1 + (-1.44e13 - 2.50e13i)T + (-2.28e26 + 3.95e26i)T^{2} \) |
| 41 | \( 1 - 1.35e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 5.19e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + (3.11e13 + 5.38e13i)T + (-1.33e28 + 2.30e28i)T^{2} \) |
| 53 | \( 1 + (-5.06e14 - 2.92e14i)T + (1.02e29 + 1.77e29i)T^{2} \) |
| 59 | \( 1 + (-6.57e14 + 1.13e15i)T + (-6.35e29 - 1.10e30i)T^{2} \) |
| 61 | \( 1 + (8.32e14 - 4.80e14i)T + (1.12e30 - 1.94e30i)T^{2} \) |
| 67 | \( 1 + (-1.79e15 + 3.10e15i)T + (-5.52e30 - 9.56e30i)T^{2} \) |
| 71 | \( 1 + 1.02e16iT - 2.96e31T^{2} \) |
| 73 | \( 1 + (-5.87e14 - 3.38e14i)T + (2.37e31 + 4.11e31i)T^{2} \) |
| 79 | \( 1 + (1.04e16 + 1.81e16i)T + (-9.09e31 + 1.57e32i)T^{2} \) |
| 83 | \( 1 - 7.37e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + (-8.89e15 - 1.53e16i)T + (-6.89e32 + 1.19e33i)T^{2} \) |
| 97 | \( 1 + 6.98e16iT - 5.95e33T^{2} \) |
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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
−14.18819319785020892578470961699, −13.27015182885208278473324108307, −11.86779850662772524309710435532, −10.80244202149561247042897004126, −8.880442567095465932647051299105, −8.015336127342820224964306911257, −5.20585134608729277209721206769, −4.44230409979607902913080489391, −3.13551175244671144667291019275, −1.79652819972589592904045397448,
0.75963465594876695188312861813, 2.57924765504726970050808640436, 4.01325354851947968244808176505, 5.51675528521579713498266329244, 7.15396204484886543627012222435, 7.88500206246775454606196940462, 9.917529181330786710560105199633, 11.70485705207244855766218741051, 13.12254217868163718212273514314, 14.04129101183501244964472807123
