| L(s) = 1 | + (52.0 − 30.0i)2-s + (−5.71e3 − 9.82e3i)3-s + (−6.37e4 + 1.10e5i)4-s + (−3.68e5 − 6.38e5i)5-s + (−5.92e5 − 3.39e5i)6-s + (−1.19e7 − 9.53e6i)7-s + 1.55e7i·8-s + (−6.39e7 + 1.12e8i)9-s + (−3.84e7 − 2.21e7i)10-s + (−9.52e8 − 5.50e8i)11-s + (1.44e9 − 4.29e6i)12-s − 1.88e9i·13-s + (−9.06e8 − 1.38e8i)14-s + (−4.17e9 + 7.27e9i)15-s + (−7.88e9 − 1.36e10i)16-s + (−1.51e10 + 2.61e10i)17-s + ⋯ |
| L(s) = 1 | + (0.143 − 0.0830i)2-s + (−0.502 − 0.864i)3-s + (−0.486 + 0.842i)4-s + (−0.422 − 0.731i)5-s + (−0.144 − 0.0826i)6-s + (−0.780 − 0.625i)7-s + 0.327i·8-s + (−0.494 + 0.868i)9-s + (−0.121 − 0.0701i)10-s + (−1.34 − 0.773i)11-s + (0.972 − 0.00288i)12-s − 0.640i·13-s + (−0.164 − 0.0251i)14-s + (−0.420 + 0.732i)15-s + (−0.459 − 0.795i)16-s + (−0.525 + 0.910i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.3102437054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3102437054\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.71e3 + 9.82e3i)T \) |
| 7 | \( 1 + (1.19e7 + 9.53e6i)T \) |
| good | 2 | \( 1 + (-52.0 + 30.0i)T + (6.55e4 - 1.13e5i)T^{2} \) |
| 5 | \( 1 + (3.68e5 + 6.38e5i)T + (-3.81e11 + 6.60e11i)T^{2} \) |
| 11 | \( 1 + (9.52e8 + 5.50e8i)T + (2.52e17 + 4.37e17i)T^{2} \) |
| 13 | \( 1 + 1.88e9iT - 8.65e18T^{2} \) |
| 17 | \( 1 + (1.51e10 - 2.61e10i)T + (-4.13e20 - 7.16e20i)T^{2} \) |
| 19 | \( 1 + (2.25e10 - 1.29e10i)T + (2.74e21 - 4.74e21i)T^{2} \) |
| 23 | \( 1 + (2.11e11 - 1.21e11i)T + (7.05e22 - 1.22e23i)T^{2} \) |
| 29 | \( 1 + 3.59e12iT - 7.25e24T^{2} \) |
| 31 | \( 1 + (-2.63e12 - 1.51e12i)T + (1.12e25 + 1.95e25i)T^{2} \) |
| 37 | \( 1 + (-9.73e12 - 1.68e13i)T + (-2.28e26 + 3.95e26i)T^{2} \) |
| 41 | \( 1 - 2.52e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 8.69e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + (6.04e13 + 1.04e14i)T + (-1.33e28 + 2.30e28i)T^{2} \) |
| 53 | \( 1 + (-5.11e14 - 2.95e14i)T + (1.02e29 + 1.77e29i)T^{2} \) |
| 59 | \( 1 + (2.14e14 - 3.72e14i)T + (-6.35e29 - 1.10e30i)T^{2} \) |
| 61 | \( 1 + (9.36e14 - 5.40e14i)T + (1.12e30 - 1.94e30i)T^{2} \) |
| 67 | \( 1 + (7.20e14 - 1.24e15i)T + (-5.52e30 - 9.56e30i)T^{2} \) |
| 71 | \( 1 + 4.32e15iT - 2.96e31T^{2} \) |
| 73 | \( 1 + (3.79e15 + 2.19e15i)T + (2.37e31 + 4.11e31i)T^{2} \) |
| 79 | \( 1 + (1.30e16 + 2.26e16i)T + (-9.09e31 + 1.57e32i)T^{2} \) |
| 83 | \( 1 + 3.07e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + (-1.28e16 - 2.22e16i)T + (-6.89e32 + 1.19e33i)T^{2} \) |
| 97 | \( 1 - 8.07e16iT - 5.95e33T^{2} \) |
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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
−13.49895150974195054772230773519, −13.05900495476431674374867157035, −12.06054958912436950025280380415, −10.53172078553895385188296157051, −8.426658396404943242187105918787, −7.67935573394657036802504713637, −5.91174248265680342729185095697, −4.33994775321871474962241721110, −2.77222289100265136303575946254, −0.62010083729791878942260504702,
0.16283023737039486625390529413, 2.67060911961538187897752280075, 4.34294271980908956739000857739, 5.51842612316458524195919982342, 6.83635390357440434925259177817, 9.122813421011068297024642473971, 10.12148205640048631311230368191, 11.15598881569896057169083812405, 12.74942438568892577651833337039, 14.40183070200777643148020424756
