| L(s) = 1 | + (−0.909 + 1.08i)2-s + (−1.44 − 2.62i)3-s + (−0.347 − 1.96i)4-s + (0.696 − 1.91i)5-s + (4.16 + 0.824i)6-s + (2.23 − 12.6i)7-s + (2.44 + 1.41i)8-s + (−4.82 + 7.59i)9-s + (1.44 + 2.49i)10-s + (1.28 + 3.52i)11-s + (−4.67 + 3.75i)12-s + (6.69 − 5.61i)13-s + (11.6 + 13.9i)14-s + (−6.04 + 0.934i)15-s + (−3.75 + 1.36i)16-s + (−20.4 + 11.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.541i)2-s + (−0.481 − 0.876i)3-s + (−0.0868 − 0.492i)4-s + (0.139 − 0.382i)5-s + (0.693 + 0.137i)6-s + (0.318 − 1.80i)7-s + (0.306 + 0.176i)8-s + (−0.535 + 0.844i)9-s + (0.144 + 0.249i)10-s + (0.116 + 0.320i)11-s + (−0.389 + 0.313i)12-s + (0.515 − 0.432i)13-s + (0.834 + 0.994i)14-s + (−0.402 + 0.0623i)15-s + (−0.234 + 0.0855i)16-s + (−1.20 + 0.694i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.663791 - 0.438926i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.663791 - 0.438926i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.909 - 1.08i)T \) |
| 3 | \( 1 + (1.44 + 2.62i)T \) |
| good | 5 | \( 1 + (-0.696 + 1.91i)T + (-19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-2.23 + 12.6i)T + (-46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-1.28 - 3.52i)T + (-92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-6.69 + 5.61i)T + (29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (20.4 - 11.8i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (2.58 - 4.47i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-34.3 + 6.04i)T + (497. - 180. i)T^{2} \) |
| 29 | \( 1 + (-18.1 + 21.6i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (0.600 + 3.40i)T + (-903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-27.7 - 48.1i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (10.1 + 12.1i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-49.2 + 17.9i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (53.2 + 9.38i)T + (2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 - 0.286iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-3.64 + 10.0i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (16.4 - 93.1i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-16.2 + 13.6i)T + (779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (89.4 - 51.6i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-11.9 + 20.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (9.54 + 8.00i)T + (1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-46.6 + 55.5i)T + (-1.19e3 - 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-56.5 - 32.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (31.7 - 11.5i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−14.90363429917253590143454441365, −13.56565974373319547601128741913, −13.01687372066844952527320525213, −11.20183071634591368519721816793, −10.38561097329307145076780886716, −8.561597079264373521874578031933, −7.40448259825040489315453313532, −6.44537674758455699603618558997, −4.66194711753177491513386636002, −1.08079542104223840868127598271,
2.80582066782917111834470299874, 4.90079029174919698940774443217, 6.40674332647220315642666910005, 8.775662894295150828143879408952, 9.262814074472917540702068752394, 10.94544763110810877905987022562, 11.47111924146167859733378686067, 12.69003592896833604905522009081, 14.44898612145760808844804441730, 15.50241018116567217989551588433
