| L(s) = 1 | + (14.6 + 2.57i)2-s + (19.5 − 18.6i)3-s + (146. + 53.3i)4-s + (−47.4 − 56.5i)5-s + (333. − 222. i)6-s + (−559. + 203. i)7-s + (1.17e3 + 681. i)8-s + (33.6 − 728. i)9-s + (−547. − 948. i)10-s + (−103. + 123. i)11-s + (3.85e3 − 1.69e3i)12-s + (611. + 3.47e3i)13-s + (−8.69e3 + 1.53e3i)14-s + (−1.98e3 − 219. i)15-s + (7.83e3 + 6.57e3i)16-s + (991. − 572. i)17-s + ⋯ |
| L(s) = 1 | + (1.82 + 0.321i)2-s + (0.723 − 0.690i)3-s + (2.28 + 0.832i)4-s + (−0.379 − 0.452i)5-s + (1.54 − 1.02i)6-s + (−1.63 + 0.593i)7-s + (2.30 + 1.33i)8-s + (0.0461 − 0.998i)9-s + (−0.547 − 0.948i)10-s + (−0.0777 + 0.0927i)11-s + (2.23 − 0.978i)12-s + (0.278 + 1.57i)13-s + (−3.16 + 0.558i)14-s + (−0.587 − 0.0650i)15-s + (1.91 + 1.60i)16-s + (0.201 − 0.116i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0401i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.999 + 0.0401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(4.43979 - 0.0890830i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.43979 - 0.0890830i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-19.5 + 18.6i)T \) |
| good | 2 | \( 1 + (-14.6 - 2.57i)T + (60.1 + 21.8i)T^{2} \) |
| 5 | \( 1 + (47.4 + 56.5i)T + (-2.71e3 + 1.53e4i)T^{2} \) |
| 7 | \( 1 + (559. - 203. i)T + (9.01e4 - 7.56e4i)T^{2} \) |
| 11 | \( 1 + (103. - 123. i)T + (-3.07e5 - 1.74e6i)T^{2} \) |
| 13 | \( 1 + (-611. - 3.47e3i)T + (-4.53e6 + 1.65e6i)T^{2} \) |
| 17 | \( 1 + (-991. + 572. i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-1.96e3 + 3.40e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-2.40e3 + 6.61e3i)T + (-1.13e8 - 9.51e7i)T^{2} \) |
| 29 | \( 1 + (1.72e4 + 3.03e3i)T + (5.58e8 + 2.03e8i)T^{2} \) |
| 31 | \( 1 + (1.17e4 + 4.28e3i)T + (6.79e8 + 5.70e8i)T^{2} \) |
| 37 | \( 1 + (-2.07e3 - 3.58e3i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-8.70e4 + 1.53e4i)T + (4.46e9 - 1.62e9i)T^{2} \) |
| 43 | \( 1 + (3.13e4 + 2.63e4i)T + (1.09e9 + 6.22e9i)T^{2} \) |
| 47 | \( 1 + (-4.14e4 - 1.13e5i)T + (-8.25e9 + 6.92e9i)T^{2} \) |
| 53 | \( 1 - 1.38e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-2.33e5 - 2.78e5i)T + (-7.32e9 + 4.15e10i)T^{2} \) |
| 61 | \( 1 + (3.60e4 - 1.31e4i)T + (3.94e10 - 3.31e10i)T^{2} \) |
| 67 | \( 1 + (4.40e4 + 2.50e5i)T + (-8.50e10 + 3.09e10i)T^{2} \) |
| 71 | \( 1 + (1.81e5 - 1.04e5i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (2.80e5 - 4.85e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-7.48e4 + 4.24e5i)T + (-2.28e11 - 8.31e10i)T^{2} \) |
| 83 | \( 1 + (5.81e5 + 1.02e5i)T + (3.07e11 + 1.11e11i)T^{2} \) |
| 89 | \( 1 + (-8.16e5 - 4.71e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (6.63e5 + 5.57e5i)T + (1.44e11 + 8.20e11i)T^{2} \) |
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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
−15.75527699364366517855103436201, −14.52134645268664717056666740641, −13.42222581099657697769376154258, −12.63171926138038290488803804369, −11.80329151381962527107348081513, −9.104380386912884757064743179366, −7.13401256171060305582216384359, −6.13000647349209486517433324205, −4.02483733053176619222699028630, −2.61489166470352852239593422907,
3.10480288791591119539622765628, 3.68032525158404575150801351081, 5.65114145869557750139161171283, 7.34407936539020735010582395144, 9.984179975264222495084097376660, 10.96675212894358199962866463472, 12.79134895385038050639609398312, 13.43500616482057824497518022876, 14.72502531491434389987822525768, 15.57619965044432988017058541547
