| L(s) = 1 | + (−5.85 + 10.1i)2-s + (1.63e4 + 2.82e4i)4-s + (7.52e4 + 1.30e5i)5-s + (1.21e6 − 2.10e6i)7-s − 7.65e5·8-s − 1.76e6·10-s + (5.22e7 − 9.05e7i)11-s + (−1.91e8 − 3.31e8i)13-s + (1.42e7 + 2.46e7i)14-s + (−5.30e8 + 9.18e8i)16-s + 2.61e9·17-s − 1.75e9·19-s + (−2.45e9 + 4.25e9i)20-s + (6.11e8 + 1.05e9i)22-s + (−5.57e9 − 9.66e9i)23-s + ⋯ |
| L(s) = 1 | + (−0.0323 + 0.0559i)2-s + (0.497 + 0.862i)4-s + (0.430 + 0.746i)5-s + (0.558 − 0.966i)7-s − 0.129·8-s − 0.0557·10-s + (0.808 − 1.40i)11-s + (−0.844 − 1.46i)13-s + (0.0360 + 0.0624i)14-s + (−0.493 + 0.855i)16-s + 1.54·17-s − 0.449·19-s + (−0.429 + 0.743i)20-s + (0.0522 + 0.0905i)22-s + (−0.341 − 0.591i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(2.601046633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.601046633\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 + (5.85 - 10.1i)T + (-1.63e4 - 2.83e4i)T^{2} \) |
| 5 | \( 1 + (-7.52e4 - 1.30e5i)T + (-1.52e10 + 2.64e10i)T^{2} \) |
| 7 | \( 1 + (-1.21e6 + 2.10e6i)T + (-2.37e12 - 4.11e12i)T^{2} \) |
| 11 | \( 1 + (-5.22e7 + 9.05e7i)T + (-2.08e15 - 3.61e15i)T^{2} \) |
| 13 | \( 1 + (1.91e8 + 3.31e8i)T + (-2.55e16 + 4.43e16i)T^{2} \) |
| 17 | \( 1 - 2.61e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 1.75e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + (5.57e9 + 9.66e9i)T + (-1.33e20 + 2.30e20i)T^{2} \) |
| 29 | \( 1 + (1.45e10 - 2.52e10i)T + (-4.31e21 - 7.47e21i)T^{2} \) |
| 31 | \( 1 + (6.88e10 + 1.19e11i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 - 2.67e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + (-3.82e11 - 6.63e11i)T + (-7.77e23 + 1.34e24i)T^{2} \) |
| 43 | \( 1 + (-9.27e11 + 1.60e12i)T + (-1.58e24 - 2.75e24i)T^{2} \) |
| 47 | \( 1 + (-3.52e11 + 6.09e11i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 - 7.69e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + (-6.90e12 - 1.19e13i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (-3.76e12 + 6.52e12i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (5.51e12 + 9.56e12i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 + 3.24e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 6.93e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + (9.79e13 - 1.69e14i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 + (1.96e14 - 3.40e14i)T + (-3.05e28 - 5.29e28i)T^{2} \) |
| 89 | \( 1 + 1.24e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + (1.03e13 - 1.79e13i)T + (-3.16e29 - 5.48e29i)T^{2} \) |
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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
−13.93238412735688059663719178596, −12.51429208985962407204510696256, −11.17449155282714168738642135209, −10.23705889538493817847505430945, −8.248485082572579585606286264516, −7.28221980064866939221118613088, −5.88598999776488532894682237017, −3.77314882876271531208468082564, −2.67931907607819835754100909677, −0.793216360254484738612995526513,
1.41691573601541615289206149543, 2.08840825627055683780402636808, 4.64302125003954550698053355781, 5.69751037807766437193975745155, 7.18209826581204327650841007228, 9.132037338136837769231771272104, 9.837521278293169979351744514927, 11.66926358446852038051655074179, 12.37086313585722111683702786350, 14.35233203689777931440941895924
