| L(s) = 1 | + (0.123 + 2.11i)2-s + (−2.45 + 0.287i)4-s + (3.78 + 0.897i)5-s + (−1.99 + 2.67i)7-s + (−0.175 − 0.992i)8-s + (−1.43 + 8.11i)10-s + (−0.519 − 0.550i)11-s + (2.22 − 1.11i)13-s + (−5.89 − 3.87i)14-s + (−2.74 + 0.650i)16-s + (−0.700 + 0.255i)17-s + (4.21 + 1.53i)19-s + (−9.57 − 1.11i)20-s + (1.09 − 1.16i)22-s + (−1.36 − 1.83i)23-s + ⋯ |
| L(s) = 1 | + (0.0869 + 1.49i)2-s + (−1.22 + 0.143i)4-s + (1.69 + 0.401i)5-s + (−0.752 + 1.01i)7-s + (−0.0618 − 0.350i)8-s + (−0.452 + 2.56i)10-s + (−0.156 − 0.165i)11-s + (0.616 − 0.309i)13-s + (−1.57 − 1.03i)14-s + (−0.686 + 0.162i)16-s + (−0.170 + 0.0618i)17-s + (0.967 + 0.352i)19-s + (−2.14 − 0.250i)20-s + (0.234 − 0.248i)22-s + (−0.284 − 0.382i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.195075 + 1.86396i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.195075 + 1.86396i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.123 - 2.11i)T + (-1.98 + 0.232i)T^{2} \) |
| 5 | \( 1 + (-3.78 - 0.897i)T + (4.46 + 2.24i)T^{2} \) |
| 7 | \( 1 + (1.99 - 2.67i)T + (-2.00 - 6.70i)T^{2} \) |
| 11 | \( 1 + (0.519 + 0.550i)T + (-0.639 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.22 + 1.11i)T + (7.76 - 10.4i)T^{2} \) |
| 17 | \( 1 + (0.700 - 0.255i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-4.21 - 1.53i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (1.36 + 1.83i)T + (-6.59 + 22.0i)T^{2} \) |
| 29 | \( 1 + (0.387 - 0.254i)T + (11.4 - 26.6i)T^{2} \) |
| 31 | \( 1 + (1.56 - 3.61i)T + (-21.2 - 22.5i)T^{2} \) |
| 37 | \( 1 + (3.64 + 3.05i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (0.284 - 4.88i)T + (-40.7 - 4.75i)T^{2} \) |
| 43 | \( 1 + (1.70 - 5.69i)T + (-35.9 - 23.6i)T^{2} \) |
| 47 | \( 1 + (4.26 + 9.88i)T + (-32.2 + 34.1i)T^{2} \) |
| 53 | \( 1 + (-5.75 + 9.96i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.84 + 3.01i)T + (-3.43 - 58.9i)T^{2} \) |
| 61 | \( 1 + (0.265 + 0.0310i)T + (59.3 + 14.0i)T^{2} \) |
| 67 | \( 1 + (-1.60 - 1.05i)T + (26.5 + 61.5i)T^{2} \) |
| 71 | \( 1 + (-1.17 + 6.65i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.37 - 7.81i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.250 + 4.30i)T + (-78.4 + 9.17i)T^{2} \) |
| 83 | \( 1 + (-0.148 - 2.54i)T + (-82.4 + 9.63i)T^{2} \) |
| 89 | \( 1 + (0.935 + 5.30i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-9.23 + 2.18i)T + (86.6 - 43.5i)T^{2} \) |
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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
−10.44783362396210246373623577725, −9.670846590267708554705251804612, −8.970432613637300057836002349964, −8.217031974783976231478112541299, −6.92377489921881511923970372904, −6.33059549666291263975493422832, −5.66889332466670025379956745458, −5.16026547998809094558743246651, −3.25124963925757384211018134988, −2.05669370786014956377005044437,
0.988135310052486910162050124043, 2.01111655350608730439450894633, 3.13435098238633583362433283756, 4.16852788505364815549690830661, 5.30212555330446423538575798997, 6.33158455685325855478909803829, 7.26238587370195579332938252560, 8.899924992659464526940312065178, 9.505005459437083637617323697636, 10.09665540656645200490824061448
