| L(s) = 1 | + (−0.515 + 2.92i)2-s + (−8.95 − 12.7i)3-s + (21.7 + 7.92i)4-s + (−68.6 + 57.6i)5-s + (41.9 − 19.6i)6-s + (−72.2 + 26.3i)7-s + (−81.9 + 141. i)8-s + (−82.4 + 228. i)9-s + (−133. − 230. i)10-s + (−343. − 288. i)11-s + (−93.9 − 348. i)12-s + (50.1 + 284. i)13-s + (−39.6 − 224. i)14-s + (1.35e3 + 359. i)15-s + (195. + 163. i)16-s + (642. + 1.11e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.0911 + 0.517i)2-s + (−0.574 − 0.818i)3-s + (0.680 + 0.247i)4-s + (−1.22 + 1.03i)5-s + (0.475 − 0.222i)6-s + (−0.557 + 0.202i)7-s + (−0.452 + 0.784i)8-s + (−0.339 + 0.940i)9-s + (−0.420 − 0.729i)10-s + (−0.855 − 0.717i)11-s + (−0.188 − 0.699i)12-s + (0.0823 + 0.466i)13-s + (−0.0540 − 0.306i)14-s + (1.54 + 0.412i)15-s + (0.190 + 0.159i)16-s + (0.539 + 0.933i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.471i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.141435 + 0.565142i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.141435 + 0.565142i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (8.95 + 12.7i)T \) |
| good | 2 | \( 1 + (0.515 - 2.92i)T + (-30.0 - 10.9i)T^{2} \) |
| 5 | \( 1 + (68.6 - 57.6i)T + (542. - 3.07e3i)T^{2} \) |
| 7 | \( 1 + (72.2 - 26.3i)T + (1.28e4 - 1.08e4i)T^{2} \) |
| 11 | \( 1 + (343. + 288. i)T + (2.79e4 + 1.58e5i)T^{2} \) |
| 13 | \( 1 + (-50.1 - 284. i)T + (-3.48e5 + 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-642. - 1.11e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-829. + 1.43e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (407. + 148. i)T + (4.93e6 + 4.13e6i)T^{2} \) |
| 29 | \( 1 + (291. - 1.65e3i)T + (-1.92e7 - 7.01e6i)T^{2} \) |
| 31 | \( 1 + (7.87e3 + 2.86e3i)T + (2.19e7 + 1.84e7i)T^{2} \) |
| 37 | \( 1 + (-4.42e3 - 7.66e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-321. - 1.82e3i)T + (-1.08e8 + 3.96e7i)T^{2} \) |
| 43 | \( 1 + (-1.62e4 - 1.36e4i)T + (2.55e7 + 1.44e8i)T^{2} \) |
| 47 | \( 1 + (2.10e4 - 7.67e3i)T + (1.75e8 - 1.47e8i)T^{2} \) |
| 53 | \( 1 - 2.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-2.19e4 + 1.84e4i)T + (1.24e8 - 7.04e8i)T^{2} \) |
| 61 | \( 1 + (5.14e4 - 1.87e4i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-6.34e3 - 3.59e4i)T + (-1.26e9 + 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-8.69e3 - 1.50e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-6.65e3 + 1.15e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.23e4 - 6.98e4i)T + (-2.89e9 - 1.05e9i)T^{2} \) |
| 83 | \( 1 + (-5.94e3 + 3.37e4i)T + (-3.70e9 - 1.34e9i)T^{2} \) |
| 89 | \( 1 + (-5.75e4 + 9.96e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (4.42e4 + 3.71e4i)T + (1.49e9 + 8.45e9i)T^{2} \) |
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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
−16.54714453773386618204737435209, −15.82001038456211992270804009156, −14.62258640894863069960885857767, −12.86713359222765641316882742946, −11.56312952378616340345593378073, −10.90118317307444266624043838164, −8.076833261450141590096781026918, −7.18129587185938218552723705639, −6.03033551422289563966427996867, −2.96937308634909386230539112650,
0.40152723977565310597273465462, 3.62109580607140917734041876796, 5.36136481233902475518076715579, 7.50618049555551463038681217039, 9.484578754920586741229507759222, 10.63531877134924786525665656788, 11.90856583220524287952392094661, 12.59947545622171595027744540322, 15.06420872685844789872268902525, 16.07247367207255966419520628090
