| L(s) = 1 | + (0.777 + 0.629i)2-s + (−1.57 + 0.710i)3-s + (0.207 + 0.978i)4-s + (−1.88 − 2.09i)5-s + (−1.67 − 0.442i)6-s + (−1.57 − 0.604i)7-s + (−0.453 + 0.891i)8-s + (1.99 − 2.24i)9-s + (−0.147 − 2.81i)10-s + (0.175 + 0.175i)11-s + (−1.02 − 1.39i)12-s + (2.62 − 4.54i)13-s + (−0.843 − 1.46i)14-s + (4.45 + 1.96i)15-s + (−0.913 + 0.406i)16-s + (−6.65 − 4.32i)17-s + ⋯ |
| L(s) = 1 | + (0.549 + 0.444i)2-s + (−0.912 + 0.410i)3-s + (0.103 + 0.489i)4-s + (−0.842 − 0.935i)5-s + (−0.683 − 0.180i)6-s + (−0.595 − 0.228i)7-s + (−0.160 + 0.315i)8-s + (0.663 − 0.747i)9-s + (−0.0465 − 0.888i)10-s + (0.0530 + 0.0530i)11-s + (−0.295 − 0.403i)12-s + (0.727 − 1.26i)13-s + (−0.225 − 0.390i)14-s + (1.15 + 0.507i)15-s + (−0.228 + 0.101i)16-s + (−1.61 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.516098 - 0.442197i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.516098 - 0.442197i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.777 - 0.629i)T \) |
| 3 | \( 1 + (1.57 - 0.710i)T \) |
| 61 | \( 1 + (4.59 - 6.31i)T \) |
| good | 5 | \( 1 + (1.88 + 2.09i)T + (-0.522 + 4.97i)T^{2} \) |
| 7 | \( 1 + (1.57 + 0.604i)T + (5.20 + 4.68i)T^{2} \) |
| 11 | \( 1 + (-0.175 - 0.175i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.62 + 4.54i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (6.65 + 4.32i)T + (6.91 + 15.5i)T^{2} \) |
| 19 | \( 1 + (-0.851 + 1.91i)T + (-12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (0.901 + 1.76i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (0.913 - 3.40i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-0.526 - 0.650i)T + (-6.44 + 30.3i)T^{2} \) |
| 37 | \( 1 + (-0.916 + 0.145i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (0.400 + 0.291i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (10.0 - 6.49i)T + (17.4 - 39.2i)T^{2} \) |
| 47 | \( 1 + (-9.85 + 5.68i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.54 + 2.31i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-2.64 + 3.27i)T + (-12.2 - 57.7i)T^{2} \) |
| 67 | \( 1 + (6.36 + 0.333i)T + (66.6 + 7.00i)T^{2} \) |
| 71 | \( 1 + (6.93 - 0.363i)T + (70.6 - 7.42i)T^{2} \) |
| 73 | \( 1 + (-4.23 + 4.70i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-4.53 - 6.98i)T + (-32.1 + 72.1i)T^{2} \) |
| 83 | \( 1 + (-6.69 + 0.703i)T + (81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-15.6 - 2.47i)T + (84.6 + 27.5i)T^{2} \) |
| 97 | \( 1 + (4.91 + 0.517i)T + (94.8 + 20.1i)T^{2} \) |
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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
−11.38497430209975251564935668519, −10.51570503946887000773828450079, −9.265053375857781917303246818937, −8.386919754490742805397968736506, −7.18082144789830556845204032482, −6.30316926077248050429366151524, −5.13608951447339229075933261336, −4.45440085912777035163426987618, −3.35148930529748266538219609975, −0.42452634323614201696764437360,
1.97767601870888099339156182516, 3.60877721206307389628610201910, 4.47788243362860638228840336573, 6.12008934922138628073917747700, 6.51437811104845355404935814089, 7.54648038904767265611778354445, 8.941435146999784169278340631479, 10.23863885970176980460084741664, 11.07170160103294976384124806299, 11.51863739020644623369609530885
