| L(s) = 1 | + (164. − 94.7i)2-s + (9.78e3 − 1.69e4i)4-s + (−9.37e4 − 5.41e4i)5-s + (5.84e5 + 1.01e6i)7-s − 6.02e5i·8-s − 2.05e7·10-s + (−1.28e7 + 7.43e6i)11-s + (−3.40e7 + 5.89e7i)13-s + (1.91e8 + 1.10e8i)14-s + (1.03e8 + 1.78e8i)16-s − 1.87e8i·17-s + 1.20e9·19-s + (−1.83e9 + 1.05e9i)20-s + (−1.40e9 + 2.44e9i)22-s + (1.17e9 + 6.75e8i)23-s + ⋯ |
| L(s) = 1 | + (1.28 − 0.740i)2-s + (0.597 − 1.03i)4-s + (−1.19 − 0.692i)5-s + (0.709 + 1.22i)7-s − 0.287i·8-s − 2.05·10-s + (−0.660 + 0.381i)11-s + (−0.542 + 0.939i)13-s + (1.82 + 1.05i)14-s + (0.384 + 0.665i)16-s − 0.457i·17-s + 1.34·19-s + (−1.43 + 0.827i)20-s + (−0.564 + 0.978i)22-s + (0.343 + 0.198i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(2.371391483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.371391483\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-164. + 94.7i)T + (8.19e3 - 1.41e4i)T^{2} \) |
| 5 | \( 1 + (9.37e4 + 5.41e4i)T + (3.05e9 + 5.28e9i)T^{2} \) |
| 7 | \( 1 + (-5.84e5 - 1.01e6i)T + (-3.39e11 + 5.87e11i)T^{2} \) |
| 11 | \( 1 + (1.28e7 - 7.43e6i)T + (1.89e14 - 3.28e14i)T^{2} \) |
| 13 | \( 1 + (3.40e7 - 5.89e7i)T + (-1.96e15 - 3.40e15i)T^{2} \) |
| 17 | \( 1 + 1.87e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 1.20e9T + 7.99e17T^{2} \) |
| 23 | \( 1 + (-1.17e9 - 6.75e8i)T + (5.79e18 + 1.00e19i)T^{2} \) |
| 29 | \( 1 + (1.46e10 - 8.43e9i)T + (1.48e20 - 2.57e20i)T^{2} \) |
| 31 | \( 1 + (1.75e10 - 3.04e10i)T + (-3.78e20 - 6.55e20i)T^{2} \) |
| 37 | \( 1 + 9.44e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + (-1.71e11 - 9.90e10i)T + (1.89e22 + 3.28e22i)T^{2} \) |
| 43 | \( 1 + (-1.04e11 - 1.81e11i)T + (-3.69e22 + 6.39e22i)T^{2} \) |
| 47 | \( 1 + (2.57e11 - 1.48e11i)T + (1.28e23 - 2.22e23i)T^{2} \) |
| 53 | \( 1 - 3.41e11iT - 1.37e24T^{2} \) |
| 59 | \( 1 + (1.82e12 + 1.05e12i)T + (3.09e24 + 5.36e24i)T^{2} \) |
| 61 | \( 1 + (1.55e12 + 2.69e12i)T + (-4.93e24 + 8.55e24i)T^{2} \) |
| 67 | \( 1 + (3.94e12 - 6.82e12i)T + (-1.83e25 - 3.18e25i)T^{2} \) |
| 71 | \( 1 + 1.64e13iT - 8.27e25T^{2} \) |
| 73 | \( 1 + 4.02e12T + 1.22e26T^{2} \) |
| 79 | \( 1 + (6.19e12 + 1.07e13i)T + (-1.84e26 + 3.19e26i)T^{2} \) |
| 83 | \( 1 + (1.44e13 - 8.36e12i)T + (3.68e26 - 6.37e26i)T^{2} \) |
| 89 | \( 1 - 6.17e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (-4.70e13 - 8.14e13i)T + (-3.26e27 + 5.65e27i)T^{2} \) |
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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
−14.18746140096851073205042849290, −12.64266025571280934003053013984, −11.97064270619381171367795340763, −11.23680203797933329412573011756, −9.061446565559245164131414559757, −7.64807162486585918747003916643, −5.29330020096428362733503942205, −4.63513503886894091904769607833, −3.10762980545561345524518926444, −1.69068415165790454812886460499,
0.46125089921589189815408323525, 3.21501078979456774548833224664, 4.17253157005125424168240610943, 5.49418910071904487582139691068, 7.36709371401306021235248150991, 7.67366232270210037769796825134, 10.45641530283182681927791392229, 11.57903293388425686228591501658, 13.00549282802076893470789927646, 14.10809064903192510374857185641
