| L(s) = 1 | + (5.49 + 0.577i)3-s + (−1.34 + 2.32i)5-s + (0.977 − 0.207i)7-s + (21.0 + 4.47i)9-s + (−7.92 + 7.13i)11-s + (−8.95 − 20.1i)13-s + (−8.71 + 12.0i)15-s + (2.30 + 2.07i)17-s + (−15.3 − 6.82i)19-s + (5.48 − 0.576i)21-s + (30.3 − 9.84i)23-s + (8.89 + 15.4i)25-s + (65.6 + 21.3i)27-s + (−19.2 − 26.4i)29-s + (−30.7 + 3.87i)31-s + ⋯ |
| L(s) = 1 | + (1.83 + 0.192i)3-s + (−0.268 + 0.465i)5-s + (0.139 − 0.0296i)7-s + (2.33 + 0.496i)9-s + (−0.720 + 0.648i)11-s + (−0.688 − 1.54i)13-s + (−0.581 + 0.800i)15-s + (0.135 + 0.122i)17-s + (−0.806 − 0.359i)19-s + (0.261 − 0.0274i)21-s + (1.31 − 0.428i)23-s + (0.355 + 0.616i)25-s + (2.43 + 0.790i)27-s + (−0.662 − 0.911i)29-s + (−0.992 + 0.125i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.20809 + 0.369935i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.20809 + 0.369935i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (30.7 - 3.87i)T \) |
| good | 3 | \( 1 + (-5.49 - 0.577i)T + (8.80 + 1.87i)T^{2} \) |
| 5 | \( 1 + (1.34 - 2.32i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-0.977 + 0.207i)T + (44.7 - 19.9i)T^{2} \) |
| 11 | \( 1 + (7.92 - 7.13i)T + (12.6 - 120. i)T^{2} \) |
| 13 | \( 1 + (8.95 + 20.1i)T + (-113. + 125. i)T^{2} \) |
| 17 | \( 1 + (-2.30 - 2.07i)T + (30.2 + 287. i)T^{2} \) |
| 19 | \( 1 + (15.3 + 6.82i)T + (241. + 268. i)T^{2} \) |
| 23 | \( 1 + (-30.3 + 9.84i)T + (427. - 310. i)T^{2} \) |
| 29 | \( 1 + (19.2 + 26.4i)T + (-259. + 799. i)T^{2} \) |
| 37 | \( 1 + (21.9 - 12.6i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-3.34 - 31.8i)T + (-1.64e3 + 349. i)T^{2} \) |
| 43 | \( 1 + (-25.5 + 57.4i)T + (-1.23e3 - 1.37e3i)T^{2} \) |
| 47 | \( 1 + (36.2 + 26.3i)T + (682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (4.73 - 22.2i)T + (-2.56e3 - 1.14e3i)T^{2} \) |
| 59 | \( 1 + (6.73 - 64.0i)T + (-3.40e3 - 723. i)T^{2} \) |
| 61 | \( 1 + 0.326iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-16.2 + 28.1i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-95.1 - 20.2i)T + (4.60e3 + 2.05e3i)T^{2} \) |
| 73 | \( 1 + (62.3 - 56.1i)T + (557. - 5.29e3i)T^{2} \) |
| 79 | \( 1 + (-17.9 - 16.1i)T + (652. + 6.20e3i)T^{2} \) |
| 83 | \( 1 + (-134. + 14.1i)T + (6.73e3 - 1.43e3i)T^{2} \) |
| 89 | \( 1 + (36.3 + 11.8i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-49.6 + 152. i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−13.16081757748356459808148226141, −12.74137554756972748684521964570, −10.79411630676510963196335491546, −9.993986014787787012317624261886, −8.882517157130021636174776489441, −7.84440864771863478907601775326, −7.19838731082693012138022810498, −4.92622443586132166980161782789, −3.40334255847419877204761969587, −2.39883377930134643333496507638,
1.96737690551873729391894262866, 3.40830240016602945989415416616, 4.74236097812085646569795824190, 6.92223077004661486681487986230, 7.926760313918925899246367463646, 8.842650025089760009729220993800, 9.494123060589336462580060050605, 10.98466065308690469443854431307, 12.50570466113615612208700657954, 13.18603762214189641936020228115
