| L(s) = 1 | + (5.36 − 1.95i)2-s + (9.76 − 12.1i)3-s + (0.410 − 0.344i)4-s + (−14.4 − 82.0i)5-s + (28.6 − 84.1i)6-s + (131. + 110. i)7-s + (−89.7 + 155. i)8-s + (−52.3 − 237. i)9-s + (−237. − 411. i)10-s + (−10.5 + 59.5i)11-s + (−0.178 − 8.35i)12-s + (790. + 287. i)13-s + (917. + 334. i)14-s + (−1.13e3 − 625. i)15-s + (−180. + 1.02e3i)16-s + (−173. − 300. i)17-s + ⋯ |
| L(s) = 1 | + (0.947 − 0.344i)2-s + (0.626 − 0.779i)3-s + (0.0128 − 0.0107i)4-s + (−0.258 − 1.46i)5-s + (0.324 − 0.954i)6-s + (1.01 + 0.849i)7-s + (−0.495 + 0.858i)8-s + (−0.215 − 0.976i)9-s + (−0.751 − 1.30i)10-s + (−0.0261 + 0.148i)11-s + (−0.000357 − 0.0167i)12-s + (1.29 + 0.472i)13-s + (1.25 + 0.455i)14-s + (−1.30 − 0.717i)15-s + (−0.176 + 1.00i)16-s + (−0.145 − 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.13560 - 1.48620i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13560 - 1.48620i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-9.76 + 12.1i)T \) |
| good | 2 | \( 1 + (-5.36 + 1.95i)T + (24.5 - 20.5i)T^{2} \) |
| 5 | \( 1 + (14.4 + 82.0i)T + (-2.93e3 + 1.06e3i)T^{2} \) |
| 7 | \( 1 + (-131. - 110. i)T + (2.91e3 + 1.65e4i)T^{2} \) |
| 11 | \( 1 + (10.5 - 59.5i)T + (-1.51e5 - 5.50e4i)T^{2} \) |
| 13 | \( 1 + (-790. - 287. i)T + (2.84e5 + 2.38e5i)T^{2} \) |
| 17 | \( 1 + (173. + 300. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (1.01e3 - 1.76e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.79e3 - 1.50e3i)T + (1.11e6 - 6.33e6i)T^{2} \) |
| 29 | \( 1 + (-7.23e3 + 2.63e3i)T + (1.57e7 - 1.31e7i)T^{2} \) |
| 31 | \( 1 + (2.05e3 - 1.72e3i)T + (4.97e6 - 2.81e7i)T^{2} \) |
| 37 | \( 1 + (5.49e3 + 9.50e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (8.62e3 + 3.13e3i)T + (8.87e7 + 7.44e7i)T^{2} \) |
| 43 | \( 1 + (-1.32e3 + 7.48e3i)T + (-1.38e8 - 5.02e7i)T^{2} \) |
| 47 | \( 1 + (5.30e3 + 4.44e3i)T + (3.98e7 + 2.25e8i)T^{2} \) |
| 53 | \( 1 + 1.42e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-6.58e3 - 3.73e4i)T + (-6.71e8 + 2.44e8i)T^{2} \) |
| 61 | \( 1 + (-5.32e3 - 4.46e3i)T + (1.46e8 + 8.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.55e4 - 5.66e3i)T + (1.03e9 + 8.67e8i)T^{2} \) |
| 71 | \( 1 + (-2.17e3 - 3.76e3i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + (2.41e4 - 4.18e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (2.11e4 - 7.71e3i)T + (2.35e9 - 1.97e9i)T^{2} \) |
| 83 | \( 1 + (3.41e4 - 1.24e4i)T + (3.01e9 - 2.53e9i)T^{2} \) |
| 89 | \( 1 + (-2.29e4 + 3.97e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-2.79e4 + 1.58e5i)T + (-8.06e9 - 2.93e9i)T^{2} \) |
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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
−15.77140573957084081883156754601, −14.38371992772996064400365543178, −13.43954066763273826118067237410, −12.33615521838777454190881379131, −11.76331620879500031372695654772, −8.726529926749125936529754116958, −8.332999973386835925465373763001, −5.63657566771368939454214854584, −4.08681209175927160239183943520, −1.76992438404508566455282983722,
3.31724796450057901328899833903, 4.58425121204329060209302789855, 6.59069473100184346732552300696, 8.288742540908266425803652569609, 10.33518632761062405581656720390, 11.10354019893953743535728512725, 13.47736434562879524037895074483, 14.23691253003546164131405123198, 14.99892664054468329712057196763, 15.88238360968814905844861781978
