| L(s) = 1 | + (−104. − 60.3i)2-s + (−906. − 1.57e3i)4-s + (1.00e4 − 5.77e3i)5-s + (3.08e5 − 5.34e5i)7-s + 2.19e6i·8-s − 1.39e6·10-s + (3.00e5 + 1.73e5i)11-s + (−7.94e6 − 1.37e7i)13-s + (−6.44e7 + 3.72e7i)14-s + (1.17e8 − 2.03e8i)16-s + 6.90e8i·17-s − 1.07e9·19-s + (−1.81e7 − 1.04e7i)20-s + (−2.09e7 − 3.62e7i)22-s + (−4.31e8 + 2.49e8i)23-s + ⋯ |
| L(s) = 1 | + (−0.816 − 0.471i)2-s + (−0.0553 − 0.0958i)4-s + (0.128 − 0.0739i)5-s + (0.374 − 0.648i)7-s + 1.04i·8-s − 0.139·10-s + (0.0154 + 0.00889i)11-s + (−0.126 − 0.219i)13-s + (−0.611 + 0.353i)14-s + (0.438 − 0.759i)16-s + 1.68i·17-s − 1.20·19-s + (−0.0141 − 0.00818i)20-s + (−0.00838 − 0.0145i)22-s + (−0.126 + 0.0731i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(1.041028865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.041028865\) |
| \(L(8)\) |
|
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 | \( 1 + (104. + 60.3i)T + (8.19e3 + 1.41e4i)T^{2} \) |
| 5 | \( 1 + (-1.00e4 + 5.77e3i)T + (3.05e9 - 5.28e9i)T^{2} \) |
| 7 | \( 1 + (-3.08e5 + 5.34e5i)T + (-3.39e11 - 5.87e11i)T^{2} \) |
| 11 | \( 1 + (-3.00e5 - 1.73e5i)T + (1.89e14 + 3.28e14i)T^{2} \) |
| 13 | \( 1 + (7.94e6 + 1.37e7i)T + (-1.96e15 + 3.40e15i)T^{2} \) |
| 17 | \( 1 - 6.90e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 1.07e9T + 7.99e17T^{2} \) |
| 23 | \( 1 + (4.31e8 - 2.49e8i)T + (5.79e18 - 1.00e19i)T^{2} \) |
| 29 | \( 1 + (-1.04e10 - 6.01e9i)T + (1.48e20 + 2.57e20i)T^{2} \) |
| 31 | \( 1 + (2.94e9 + 5.10e9i)T + (-3.78e20 + 6.55e20i)T^{2} \) |
| 37 | \( 1 - 1.38e11T + 9.01e21T^{2} \) |
| 41 | \( 1 + (-1.75e11 + 1.01e11i)T + (1.89e22 - 3.28e22i)T^{2} \) |
| 43 | \( 1 + (-2.34e11 + 4.06e11i)T + (-3.69e22 - 6.39e22i)T^{2} \) |
| 47 | \( 1 + (-3.07e11 - 1.77e11i)T + (1.28e23 + 2.22e23i)T^{2} \) |
| 53 | \( 1 - 9.97e11iT - 1.37e24T^{2} \) |
| 59 | \( 1 + (2.06e12 - 1.19e12i)T + (3.09e24 - 5.36e24i)T^{2} \) |
| 61 | \( 1 + (-2.15e12 + 3.74e12i)T + (-4.93e24 - 8.55e24i)T^{2} \) |
| 67 | \( 1 + (-5.72e12 - 9.90e12i)T + (-1.83e25 + 3.18e25i)T^{2} \) |
| 71 | \( 1 + 1.28e13iT - 8.27e25T^{2} \) |
| 73 | \( 1 + 4.56e12T + 1.22e26T^{2} \) |
| 79 | \( 1 + (6.34e12 - 1.09e13i)T + (-1.84e26 - 3.19e26i)T^{2} \) |
| 83 | \( 1 + (-5.10e12 - 2.94e12i)T + (3.68e26 + 6.37e26i)T^{2} \) |
| 89 | \( 1 - 1.73e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (5.09e12 - 8.83e12i)T + (-3.26e27 - 5.65e27i)T^{2} \) |
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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
−14.10165675050464730834409388748, −12.69025474089544328005320706403, −11.02717560803288030266156464075, −10.31844029132168703656486517129, −8.943974552854035518294250866465, −7.78680258017484603579116596554, −5.87898611674169516307918812393, −4.19645350755615884586455393301, −2.09514909570461958755168172936, −0.893570937766477321037077286327,
0.57159720516027241336591808298, 2.50996892214174093533900647263, 4.47359732214265480119059316373, 6.30124522806972628954290998659, 7.69142452351782994252794175869, 8.820806311381927089175127701481, 9.871352005607851753069133552990, 11.55338716015800576707325656340, 12.83841171964604541264532167890, 14.29486630397164012522611536339
