| L(s) = 1 | + (−6.01 + 3.47i)2-s + (−103. + 179. i)4-s + (−331. − 191. i)5-s + (467. + 809. i)7-s − 3.21e3i·8-s + 2.65e3·10-s + (−4.89e3 + 2.82e3i)11-s + (1.76e4 − 3.05e4i)13-s + (−5.62e3 − 3.24e3i)14-s + (−1.54e4 − 2.67e4i)16-s − 1.52e5i·17-s + 1.91e5·19-s + (6.88e4 − 3.97e4i)20-s + (1.95e4 − 3.39e4i)22-s + (−1.32e5 − 7.63e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.375 + 0.216i)2-s + (−0.405 + 0.702i)4-s + (−0.530 − 0.306i)5-s + (0.194 + 0.337i)7-s − 0.786i·8-s + 0.265·10-s + (−0.334 + 0.192i)11-s + (0.618 − 1.07i)13-s + (−0.146 − 0.0844i)14-s + (−0.235 − 0.407i)16-s − 1.83i·17-s + 1.46·19-s + (0.430 − 0.248i)20-s + (0.0836 − 0.144i)22-s + (−0.472 − 0.272i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.614604 - 0.442585i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.614604 - 0.442585i\) |
| \(L(5)\) |
|
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 | \( 1 + (6.01 - 3.47i)T + (128 - 221. i)T^{2} \) |
| 5 | \( 1 + (331. + 191. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-467. - 809. i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (4.89e3 - 2.82e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (-1.76e4 + 3.05e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 + 1.52e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.91e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + (1.32e5 + 7.63e4i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (4.01e5 - 2.31e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (3.93e5 - 6.82e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + 1.10e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-3.14e6 - 1.81e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (1.51e6 + 2.62e6i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (4.75e6 - 2.74e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + 1.41e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (7.58e6 + 4.37e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (3.47e6 + 6.01e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-7.08e6 + 1.22e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 7.97e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.61e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + (1.37e6 + 2.38e6i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (3.52e7 - 2.03e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 - 2.90e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (4.58e7 + 7.93e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−15.73496791147205758807677947858, −13.91214694158655252006097370827, −12.66066160101421145382512695486, −11.57440190799302432231540464139, −9.695888564297503815921366401407, −8.380878381649736271394777974830, −7.37066775229451780283752726069, −5.08366460466008732270719504013, −3.26269351802262015539405333360, −0.42335683278007348979099732730,
1.48681183644745952940890842516, 3.98680534333727308949029478195, 5.86803756492908380496914235900, 7.77361272646775427687731467263, 9.219590899109874912794716457531, 10.59786700496461799447333181522, 11.55990882204009451614837898376, 13.43490882060547745615975222253, 14.50126810123344750909129670409, 15.66702719326691015710589931523
