| L(s) = 1 | + (−14.5 + 6.68i)2-s + (−78.1 − 25.3i)3-s + (166. − 194. i)4-s + (−624. − 1.49i)5-s + (1.30e3 − 153. i)6-s + 3.85e3i·7-s + (−1.12e3 + 3.93e3i)8-s + (151. + 109. i)9-s + (9.09e3 − 4.15e3i)10-s + (−7.60e3 − 1.04e4i)11-s + (−1.79e4 + 1.09e4i)12-s + (3.88e4 + 2.82e4i)13-s + (−2.57e4 − 5.60e4i)14-s + (4.87e4 + 1.59e4i)15-s + (−1.00e4 − 6.47e4i)16-s + (−3.22e4 − 9.92e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.908 + 0.417i)2-s + (−0.964 − 0.313i)3-s + (0.650 − 0.759i)4-s + (−0.999 − 0.00239i)5-s + (1.00 − 0.118i)6-s + 1.60i·7-s + (−0.274 + 0.961i)8-s + (0.0230 + 0.0167i)9-s + (0.909 − 0.415i)10-s + (−0.519 − 0.715i)11-s + (−0.865 + 0.528i)12-s + (1.36 + 0.988i)13-s + (−0.671 − 1.45i)14-s + (0.963 + 0.315i)15-s + (−0.152 − 0.988i)16-s + (−0.385 − 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0369247 + 0.268173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0369247 + 0.268173i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (14.5 - 6.68i)T \) |
| 5 | \( 1 + (624. + 1.49i)T \) |
| good | 3 | \( 1 + (78.1 + 25.3i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 3.85e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (7.60e3 + 1.04e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-3.88e4 - 2.82e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (3.22e4 + 9.92e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-1.22e5 + 3.96e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-1.21e4 - 1.67e4i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (2.61e5 - 8.04e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-1.17e6 + 3.82e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (7.83e5 + 5.69e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (1.93e4 + 1.40e4i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 2.10e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.86e6 - 1.25e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-1.07e6 + 3.31e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (9.22e6 - 1.26e7i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (1.68e7 - 1.22e7i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-1.95e7 + 6.34e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-4.99e6 - 1.62e6i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (3.51e7 - 2.55e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (-3.39e7 - 1.10e7i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (4.40e7 - 1.43e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (1.11e7 - 8.10e6i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (2.89e7 - 8.92e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.14413782171799194264529688951, −11.53441397465530195588753488476, −10.99212011882926354848286905064, −9.088924090572572256758051382085, −8.563305754854787786776547380517, −7.13462744869237013878261822389, −6.07354121304906291813439414353, −5.19007041633177421261314665212, −2.86217118248917829028008631579, −1.01683083406235638035181121630,
0.17496802912258623562636998992, 1.10248592224817126887778638833, 3.39771113293464781135280968442, 4.43755137460471625801170080470, 6.29525396244874635759409370110, 7.59642241257776715414202198757, 8.260380281044267970092113257878, 10.15613227175630921708744061115, 10.67878962538904434471902103345, 11.36336977156016373938175298895
