| L(s) = 1 | + (87.7 + 152. i)2-s + (979. − 1.69e3i)4-s + (−3.17e4 + 5.49e4i)5-s + (1.89e6 + 3.29e6i)7-s + 6.09e6·8-s − 1.11e7·10-s + (−2.60e7 − 4.50e7i)11-s + (1.76e8 − 3.06e8i)13-s + (−3.33e8 + 5.77e8i)14-s + (5.02e8 + 8.70e8i)16-s + 7.31e7·17-s + 3.47e9·19-s + (6.21e7 + 1.07e8i)20-s + (4.56e9 − 7.90e9i)22-s + (−1.17e10 + 2.02e10i)23-s + ⋯ |
| L(s) = 1 | + (0.484 + 0.839i)2-s + (0.0298 − 0.0517i)4-s + (−0.181 + 0.314i)5-s + (0.871 + 1.51i)7-s + 1.02·8-s − 0.352·10-s + (−0.402 − 0.697i)11-s + (0.782 − 1.35i)13-s + (−0.845 + 1.46i)14-s + (0.468 + 0.811i)16-s + 0.0432·17-s + 0.893·19-s + (0.0108 + 0.0188i)20-s + (0.390 − 0.676i)22-s + (−0.716 + 1.24i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(3.300402792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.300402792\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 + (-87.7 - 152. i)T + (-1.63e4 + 2.83e4i)T^{2} \) |
| 5 | \( 1 + (3.17e4 - 5.49e4i)T + (-1.52e10 - 2.64e10i)T^{2} \) |
| 7 | \( 1 + (-1.89e6 - 3.29e6i)T + (-2.37e12 + 4.11e12i)T^{2} \) |
| 11 | \( 1 + (2.60e7 + 4.50e7i)T + (-2.08e15 + 3.61e15i)T^{2} \) |
| 13 | \( 1 + (-1.76e8 + 3.06e8i)T + (-2.55e16 - 4.43e16i)T^{2} \) |
| 17 | \( 1 - 7.31e7T + 2.86e18T^{2} \) |
| 19 | \( 1 - 3.47e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + (1.17e10 - 2.02e10i)T + (-1.33e20 - 2.30e20i)T^{2} \) |
| 29 | \( 1 + (-4.41e10 - 7.63e10i)T + (-4.31e21 + 7.47e21i)T^{2} \) |
| 31 | \( 1 + (5.82e10 - 1.00e11i)T + (-1.17e22 - 2.03e22i)T^{2} \) |
| 37 | \( 1 + 1.72e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + (1.41e11 - 2.45e11i)T + (-7.77e23 - 1.34e24i)T^{2} \) |
| 43 | \( 1 + (-6.98e10 - 1.21e11i)T + (-1.58e24 + 2.75e24i)T^{2} \) |
| 47 | \( 1 + (1.27e11 + 2.20e11i)T + (-6.03e24 + 1.04e25i)T^{2} \) |
| 53 | \( 1 - 1.42e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + (7.42e12 - 1.28e13i)T + (-1.82e26 - 3.16e26i)T^{2} \) |
| 61 | \( 1 + (-2.52e12 - 4.36e12i)T + (-3.01e26 + 5.21e26i)T^{2} \) |
| 67 | \( 1 + (1.04e13 - 1.81e13i)T + (-1.23e27 - 2.13e27i)T^{2} \) |
| 71 | \( 1 + 1.40e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 6.98e12T + 8.90e27T^{2} \) |
| 79 | \( 1 + (7.45e13 + 1.29e14i)T + (-1.45e28 + 2.52e28i)T^{2} \) |
| 83 | \( 1 + (2.42e14 + 4.20e14i)T + (-3.05e28 + 5.29e28i)T^{2} \) |
| 89 | \( 1 - 2.68e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + (2.39e14 + 4.14e14i)T + (-3.16e29 + 5.48e29i)T^{2} \) |
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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
−14.51006079129041709319848880019, −13.28981900996274151571541042267, −11.70282050573966622030534213995, −10.58746112968739960194554138031, −8.631125504294636600160407042736, −7.54743728999642929990623249309, −5.77337664004698003197716805714, −5.28784057134507910177677458121, −3.13820637032662639457672492582, −1.40596135440876095467177896215,
0.908407079615483483885487167952, 2.08927960280428828499026272848, 3.97420691400084084257391428557, 4.59273059149267208588319575506, 7.01562445883775510399663586435, 8.148047902580397145343429427079, 10.15141142183150489972365117286, 11.17041276796316179168766769727, 12.16076979589519171656653541229, 13.52321660152588780534794171353
