| L(s) = 1 | + (0.560 + 3.17i)3-s + (−1.93 + 1.62i)5-s + (1.93 − 3.35i)7-s + (−6.96 + 2.53i)9-s + (1.17 + 2.03i)11-s + (0.145 − 0.824i)13-s + (−6.25 − 5.25i)15-s + (0.900 + 0.327i)17-s + (4.35 + 0.0632i)19-s + (11.7 + 4.28i)21-s + (3.34 + 2.80i)23-s + (0.245 − 1.39i)25-s + (−7.11 − 12.3i)27-s + (3.25 − 1.18i)29-s + (2.31 − 4.01i)31-s + ⋯ |
| L(s) = 1 | + (0.323 + 1.83i)3-s + (−0.867 + 0.727i)5-s + (0.733 − 1.26i)7-s + (−2.32 + 0.844i)9-s + (0.353 + 0.612i)11-s + (0.0403 − 0.228i)13-s + (−1.61 − 1.35i)15-s + (0.218 + 0.0794i)17-s + (0.999 + 0.0144i)19-s + (2.56 + 0.934i)21-s + (0.697 + 0.585i)23-s + (0.0490 − 0.278i)25-s + (−1.36 − 2.37i)27-s + (0.605 − 0.220i)29-s + (0.416 − 0.721i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.667252 + 0.900691i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.667252 + 0.900691i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-4.35 - 0.0632i)T \) |
| good | 3 | \( 1 + (-0.560 - 3.17i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (1.93 - 1.62i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.93 + 3.35i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.17 - 2.03i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.145 + 0.824i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.900 - 0.327i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-3.34 - 2.80i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.25 + 1.18i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.31 + 4.01i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.22T + 37T^{2} \) |
| 41 | \( 1 + (1.63 + 9.25i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (3.42 - 2.87i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-6.73 + 2.45i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (3.24 + 2.72i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (7.92 + 2.88i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (4.89 + 4.10i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-12.7 + 4.63i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (6.53 - 5.48i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.54 - 8.77i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.647 + 3.67i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.23 + 2.13i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.396 - 2.24i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-0.166 - 0.0605i)T + (74.3 + 62.3i)T^{2} \) |
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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
−13.81020027062054880136873414123, −11.80078240223992176404943167534, −11.00748291545936435141367151961, −10.36203406261550170435953400466, −9.458998386017950943139724784716, −8.120234486119367678518246516037, −7.19023234549246917322331940599, −5.13811659221697972602746431599, −4.11487083630316234365699608302, −3.33461707007631781166032241626,
1.27680565541480310475977999137, 2.92833934409841996085641645278, 5.12223883633110212299795092473, 6.36941495451777279764713634337, 7.59583741555025952116245521914, 8.465976767063164683909598699631, 8.908652955226260928138293068524, 11.30400935490095112699782286836, 12.10972732175207549241020033326, 12.33377195608791523706800408014
