| L(s) = 1 | + (−0.974 + 1.74i)2-s + (−0.586 − 2.94i)3-s + (−2.10 − 3.40i)4-s + (−1.26 + 7.19i)5-s + (5.71 + 1.84i)6-s + (−2.67 − 3.18i)7-s + (7.99 − 0.353i)8-s + (−8.31 + 3.45i)9-s + (−11.3 − 9.23i)10-s + (−18.4 + 3.24i)11-s + (−8.78 + 8.17i)12-s + (−15.1 + 5.50i)13-s + (8.17 − 1.56i)14-s + (21.9 − 0.490i)15-s + (−7.17 + 14.3i)16-s + (7.49 − 12.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.487 + 0.873i)2-s + (−0.195 − 0.980i)3-s + (−0.525 − 0.850i)4-s + (−0.253 + 1.43i)5-s + (0.951 + 0.306i)6-s + (−0.382 − 0.455i)7-s + (0.999 − 0.0441i)8-s + (−0.923 + 0.383i)9-s + (−1.13 − 0.923i)10-s + (−1.67 + 0.295i)11-s + (−0.731 + 0.681i)12-s + (−1.16 + 0.423i)13-s + (0.584 − 0.111i)14-s + (1.46 − 0.0327i)15-s + (−0.448 + 0.893i)16-s + (0.441 − 0.763i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.154i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0130941 - 0.168862i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0130941 - 0.168862i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.974 - 1.74i)T \) |
| 3 | \( 1 + (0.586 + 2.94i)T \) |
| good | 5 | \( 1 + (1.26 - 7.19i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (2.67 + 3.18i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (18.4 - 3.24i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (15.1 - 5.50i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-7.49 + 12.9i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-0.338 + 0.195i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (10.6 - 12.7i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-36.9 - 13.4i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-14.0 + 16.6i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-5.57 + 9.65i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (56.8 - 20.6i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (24.4 - 4.31i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-14.6 - 17.4i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 51.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (82.3 + 14.5i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-19.0 + 15.9i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-17.9 - 49.4i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (75.5 + 43.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-17.8 - 30.9i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-3.59 + 9.87i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (34.4 - 94.7i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (7.29 + 12.6i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (1.45 + 8.25i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−14.07054263279029420115779819691, −13.31864384014145112406321118647, −11.88661146515179538580712560994, −10.60237687525288191098934919986, −9.889465522899868270062967073070, −7.985537260458693007451747360909, −7.30221111601300798214437242619, −6.62364091154963391653428477053, −5.18220479975669359652148970083, −2.66025384803463798176138704026,
0.14029238966193242504425531768, 2.86558464182175270881092349676, 4.54483709604087286151239593138, 5.39855108067011939944868256347, 8.072413387382374398276282811970, 8.678809345538832550362965968935, 9.929784639211855146955757471479, 10.47559204417429126279418003115, 12.06017457763442154352616227870, 12.45810478606031292568640319679
