| L(s) = 1 | + (−1.40 + 0.112i)2-s + (−2.32 + 2.01i)3-s + (1.97 − 0.317i)4-s + (−2.15 + 0.309i)5-s + (3.05 − 3.10i)6-s + (−0.974 + 2.13i)7-s + (−2.74 + 0.669i)8-s + (0.920 − 6.40i)9-s + (3.00 − 0.678i)10-s + (0.0777 − 0.264i)11-s + (−3.95 + 4.71i)12-s + (3.89 − 1.77i)13-s + (1.13 − 3.11i)14-s + (4.38 − 5.05i)15-s + (3.79 − 1.25i)16-s + (−4.80 − 3.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.996 + 0.0795i)2-s + (−1.34 + 1.16i)3-s + (0.987 − 0.158i)4-s + (−0.962 + 0.138i)5-s + (1.24 − 1.26i)6-s + (−0.368 + 0.806i)7-s + (−0.971 + 0.236i)8-s + (0.306 − 2.13i)9-s + (0.948 − 0.214i)10-s + (0.0234 − 0.0798i)11-s + (−1.14 + 1.36i)12-s + (1.08 − 0.493i)13-s + (0.303 − 0.833i)14-s + (1.13 − 1.30i)15-s + (0.949 − 0.313i)16-s + (−1.16 − 0.748i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0294405 - 0.0343057i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0294405 - 0.0343057i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 - 0.112i)T \) |
| 23 | \( 1 + (-4.77 - 0.395i)T \) |
| good | 3 | \( 1 + (2.32 - 2.01i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (2.15 - 0.309i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (0.974 - 2.13i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.0777 + 0.264i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-3.89 + 1.77i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (4.80 + 3.08i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (3.39 + 5.28i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (1.47 - 2.29i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (3.50 - 4.04i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.812 - 0.116i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (1.14 + 7.96i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (2.46 - 2.13i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + (1.58 + 0.723i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (2.72 - 1.24i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (1.80 + 1.56i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (4.38 + 14.9i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-2.63 + 0.774i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (8.30 - 5.34i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (3.77 + 8.27i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (8.83 + 1.27i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-4.27 - 4.93i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.41 + 9.87i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.74057096124682930986563964721, −11.13219597748733953765083285325, −10.66432409506805030475633808527, −9.281220782430934419159208746820, −8.714801156167959721676016079072, −7.01457634850073521358208724876, −6.09746620918027992925540906957, −4.88852750877730166884401021830, −3.31330308938096503023854653661, −0.06657092767657493051020904056,
1.53843937574697348671738852510, 4.04512874071134198091636999487, 6.08929487592850741698233487810, 6.74636599410017790756635015263, 7.68569676725411476422163608205, 8.557327620356882048955959325974, 10.25371056348786555364615298442, 11.19120091222260115366642160992, 11.53043603242470845125144057688, 12.70437435615538313693284205042
