| L(s) = 1 | + (−0.282 + 0.634i)3-s + (−1.46 + 2.53i)5-s + (−4.75 + 5.28i)7-s + (5.69 + 6.32i)9-s + (−1.43 + 6.77i)11-s + (−2.65 + 0.279i)13-s + (−1.19 − 1.64i)15-s + (1.16 + 5.45i)17-s + (−0.598 + 5.69i)19-s + (−2.00 − 4.51i)21-s + (0.00715 + 0.00232i)23-s + (8.20 + 14.2i)25-s + (−11.5 + 3.75i)27-s + (24.9 − 34.3i)29-s + (−17.2 − 25.7i)31-s + ⋯ |
| L(s) = 1 | + (−0.0941 + 0.211i)3-s + (−0.293 + 0.507i)5-s + (−0.679 + 0.755i)7-s + (0.633 + 0.703i)9-s + (−0.130 + 0.615i)11-s + (−0.204 + 0.0214i)13-s + (−0.0797 − 0.109i)15-s + (0.0682 + 0.321i)17-s + (−0.0314 + 0.299i)19-s + (−0.0956 − 0.214i)21-s + (0.000311 + 0.000101i)23-s + (0.328 + 0.568i)25-s + (−0.428 + 0.139i)27-s + (0.860 − 1.18i)29-s + (−0.555 − 0.831i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.714905 + 0.820737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.714905 + 0.820737i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (17.2 + 25.7i)T \) |
| good | 3 | \( 1 + (0.282 - 0.634i)T + (-6.02 - 6.68i)T^{2} \) |
| 5 | \( 1 + (1.46 - 2.53i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (4.75 - 5.28i)T + (-5.12 - 48.7i)T^{2} \) |
| 11 | \( 1 + (1.43 - 6.77i)T + (-110. - 49.2i)T^{2} \) |
| 13 | \( 1 + (2.65 - 0.279i)T + (165. - 35.1i)T^{2} \) |
| 17 | \( 1 + (-1.16 - 5.45i)T + (-264. + 117. i)T^{2} \) |
| 19 | \( 1 + (0.598 - 5.69i)T + (-353. - 75.0i)T^{2} \) |
| 23 | \( 1 + (-0.00715 - 0.00232i)T + (427. + 310. i)T^{2} \) |
| 29 | \( 1 + (-24.9 + 34.3i)T + (-259. - 799. i)T^{2} \) |
| 37 | \( 1 + (-24.2 + 13.9i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-49.2 + 21.9i)T + (1.12e3 - 1.24e3i)T^{2} \) |
| 43 | \( 1 + (39.8 + 4.18i)T + (1.80e3 + 384. i)T^{2} \) |
| 47 | \( 1 + (3.37 - 2.44i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-36.1 + 32.5i)T + (293. - 2.79e3i)T^{2} \) |
| 59 | \( 1 + (-32.4 - 14.4i)T + (2.32e3 + 2.58e3i)T^{2} \) |
| 61 | \( 1 - 11.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (1.39 - 2.41i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-54.2 - 60.2i)T + (-526. + 5.01e3i)T^{2} \) |
| 73 | \( 1 + (19.7 - 92.6i)T + (-4.86e3 - 2.16e3i)T^{2} \) |
| 79 | \( 1 + (-15.3 - 72.0i)T + (-5.70e3 + 2.53e3i)T^{2} \) |
| 83 | \( 1 + (0.665 + 1.49i)T + (-4.60e3 + 5.11e3i)T^{2} \) |
| 89 | \( 1 + (-31.1 + 10.1i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-13.2 - 40.6i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−13.27004056038719562325966735088, −12.48711160185854962114166582678, −11.36655860495315356861452255554, −10.25856632498301426071328104447, −9.449291504287172405786650676260, −7.983463094316129441945144411464, −6.91089050648786797367782216860, −5.58234113465606292718357174084, −4.10760755356153784915522889268, −2.43540214352421924153180488326,
0.794822893619099264168290292331, 3.35166011477278151919742750782, 4.70772761179925382269723651111, 6.37101828039126239883451347087, 7.31017546137163806532105112937, 8.650080427199088102020330204656, 9.746147103058010973364588337513, 10.78011544890036009825270162059, 12.08403469411907855341734845933, 12.83072024737743832864286488401
