| L(s) = 1 | + (6.58 − 11.4i)2-s + (13.5 − 7.79i)3-s + (−54.8 − 94.9i)4-s + (−68.9 − 39.7i)5-s − 205. i·6-s + (284. + 191. i)7-s − 601.·8-s + (121.5 − 210. i)9-s + (−908. + 524. i)10-s + (−411. − 712. i)11-s + (−1.48e3 − 854. i)12-s + 2.42e3i·13-s + (4.06e3 − 1.97e3i)14-s − 1.24e3·15-s + (−456. + 791. i)16-s + (6.75e3 − 3.89e3i)17-s + ⋯ |
| L(s) = 1 | + (0.823 − 1.42i)2-s + (0.5 − 0.288i)3-s + (−0.856 − 1.48i)4-s + (−0.551 − 0.318i)5-s − 0.951i·6-s + (0.828 + 0.559i)7-s − 1.17·8-s + (0.166 − 0.288i)9-s + (−0.908 + 0.524i)10-s + (−0.308 − 0.534i)11-s + (−0.856 − 0.494i)12-s + 1.10i·13-s + (1.48 − 0.721i)14-s − 0.367·15-s + (−0.111 + 0.193i)16-s + (1.37 − 0.793i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.01761 - 2.24836i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01761 - 2.24836i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-13.5 + 7.79i)T \) |
| 7 | \( 1 + (-284. - 191. i)T \) |
| good | 2 | \( 1 + (-6.58 + 11.4i)T + (-32 - 55.4i)T^{2} \) |
| 5 | \( 1 + (68.9 + 39.7i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (411. + 712. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 - 2.42e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-6.75e3 + 3.89e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-5.78e3 - 3.34e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (9.41e3 - 1.63e4i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 - 1.38e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (2.41e4 - 1.39e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (3.98e4 - 6.90e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 5.91e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.18e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (4.34e3 + 2.50e3i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (9.31e4 + 1.61e5i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-1.95e5 + 1.12e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.25e5 + 7.21e4i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.17e5 - 2.03e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 9.62e4T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-2.38e5 + 1.37e5i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-3.40e5 + 5.90e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 1.28e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (3.22e5 + 1.86e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 6.20e5iT - 8.32e11T^{2} \) |
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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
−16.12564156860900616670680061863, −14.43489410169520588774001579603, −13.68225643894510232533374683503, −12.03103382927762455701678215747, −11.62851439682204977024596011979, −9.736210376158986569242947480114, −8.019499986775679808237289848000, −5.12585294616413993570879270517, −3.39924450791436126773001582690, −1.55022349644524571884962061999,
3.79044233514880082465761248122, 5.31448850904758954752247635665, 7.39370344282558503386230958903, 8.127661995683913136231997827509, 10.48065601427312615298428259008, 12.53606873204608294989904815316, 13.98857684243388162070090690347, 14.82039606503355772688714888834, 15.64175250067608447735542028359, 16.89191782623276427306316603236
