| L(s) = 1 | + (37.9 − 6.69i)2-s + (156. − 186. i)3-s + (434. − 158. i)4-s + (2.07e3 − 2.47e3i)5-s + (4.68e3 − 8.11e3i)6-s + (1.99e4 + 7.26e3i)7-s + (−1.87e4 + 1.08e4i)8-s + (−1.03e4 − 5.81e4i)9-s + (6.22e4 − 1.07e5i)10-s + (8.59e3 + 1.02e4i)11-s + (3.84e4 − 1.05e5i)12-s + (6.76e4 − 3.83e5i)13-s + (8.06e5 + 1.42e5i)14-s + (−1.36e5 − 7.72e5i)15-s + (−1.00e6 + 8.40e5i)16-s + (−6.77e5 − 3.90e5i)17-s + ⋯ |
| L(s) = 1 | + (1.18 − 0.209i)2-s + (0.642 − 0.766i)3-s + (0.424 − 0.154i)4-s + (0.664 − 0.791i)5-s + (0.602 − 1.04i)6-s + (1.18 + 0.432i)7-s + (−0.571 + 0.330i)8-s + (−0.174 − 0.984i)9-s + (0.622 − 1.07i)10-s + (0.0533 + 0.0635i)11-s + (0.154 − 0.424i)12-s + (0.182 − 1.03i)13-s + (1.49 + 0.264i)14-s + (−0.179 − 1.01i)15-s + (−0.955 + 0.801i)16-s + (−0.476 − 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(3.65043 - 2.88284i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.65043 - 2.88284i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-156. + 186. i)T \) |
| good | 2 | \( 1 + (-37.9 + 6.69i)T + (962. - 350. i)T^{2} \) |
| 5 | \( 1 + (-2.07e3 + 2.47e3i)T + (-1.69e6 - 9.61e6i)T^{2} \) |
| 7 | \( 1 + (-1.99e4 - 7.26e3i)T + (2.16e8 + 1.81e8i)T^{2} \) |
| 11 | \( 1 + (-8.59e3 - 1.02e4i)T + (-4.50e9 + 2.55e10i)T^{2} \) |
| 13 | \( 1 + (-6.76e4 + 3.83e5i)T + (-1.29e11 - 4.71e10i)T^{2} \) |
| 17 | \( 1 + (6.77e5 + 3.90e5i)T + (1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (3.46e4 + 5.99e4i)T + (-3.06e12 + 5.30e12i)T^{2} \) |
| 23 | \( 1 + (-1.60e6 - 4.41e6i)T + (-3.17e13 + 2.66e13i)T^{2} \) |
| 29 | \( 1 + (-1.72e7 + 3.04e6i)T + (3.95e14 - 1.43e14i)T^{2} \) |
| 31 | \( 1 + (-6.46e6 + 2.35e6i)T + (6.27e14 - 5.26e14i)T^{2} \) |
| 37 | \( 1 + (5.78e6 - 1.00e7i)T + (-2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 + (-1.97e8 - 3.47e7i)T + (1.26e16 + 4.59e15i)T^{2} \) |
| 43 | \( 1 + (1.95e8 - 1.63e8i)T + (3.75e15 - 2.12e16i)T^{2} \) |
| 47 | \( 1 + (1.27e8 - 3.49e8i)T + (-4.02e16 - 3.38e16i)T^{2} \) |
| 53 | \( 1 - 5.55e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (3.88e8 - 4.63e8i)T + (-8.87e16 - 5.03e17i)T^{2} \) |
| 61 | \( 1 + (-1.41e9 - 5.16e8i)T + (5.46e17 + 4.58e17i)T^{2} \) |
| 67 | \( 1 + (-2.82e8 + 1.60e9i)T + (-1.71e18 - 6.23e17i)T^{2} \) |
| 71 | \( 1 + (5.15e8 + 2.97e8i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (1.82e9 + 3.15e9i)T + (-2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (-6.52e8 - 3.69e9i)T + (-8.89e18 + 3.23e18i)T^{2} \) |
| 83 | \( 1 + (2.99e9 - 5.28e8i)T + (1.45e19 - 5.30e18i)T^{2} \) |
| 89 | \( 1 + (-1.68e9 + 9.74e8i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 + (-1.09e10 + 9.21e9i)T + (1.28e19 - 7.26e19i)T^{2} \) |
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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
−14.38176505421059263409971141711, −13.40076422665927872171436827065, −12.64928840017164101913155977391, −11.48418601438609460343932156681, −9.146737504924820848307685468926, −8.047175192303865183571645264209, −5.91594549881416866498160495629, −4.72517977205694085280102746860, −2.78960094749978852388711599374, −1.35897961967160602603832375964,
2.27722989514086565491954541159, 3.92578946564884886780924774025, 5.01712042407227163779520333007, 6.70255052740931127454700370199, 8.654164025583835212490439651452, 10.19661595052022318155921852383, 11.43683611773788485735911709572, 13.38066459681031207566440066109, 14.37008324119143204520403012518, 14.58854394095468878488361377772
