| L(s) = 1 | + (1.26 + 0.629i)2-s + (0.761 − 1.83i)3-s + (1.20 + 1.59i)4-s + (0.838 − 0.427i)5-s + (2.12 − 1.84i)6-s + (−3.70 + 2.26i)7-s + (0.526 + 2.77i)8-s + (−0.674 − 0.674i)9-s + (1.33 − 0.0133i)10-s + (0.486 − 6.18i)11-s + (3.84 − 1.00i)12-s + (−2.25 + 0.542i)13-s + (−6.11 + 0.543i)14-s + (−0.146 − 1.86i)15-s + (−1.08 + 3.85i)16-s + (0.645 + 0.755i)17-s + ⋯ |
| L(s) = 1 | + (0.895 + 0.445i)2-s + (0.439 − 1.06i)3-s + (0.603 + 0.797i)4-s + (0.374 − 0.191i)5-s + (0.865 − 0.754i)6-s + (−1.39 + 0.857i)7-s + (0.186 + 0.982i)8-s + (−0.224 − 0.224i)9-s + (0.420 − 0.00421i)10-s + (0.146 − 1.86i)11-s + (1.11 − 0.290i)12-s + (−0.626 + 0.150i)13-s + (−1.63 + 0.145i)14-s + (−0.0379 − 0.481i)15-s + (−0.270 + 0.962i)16-s + (0.156 + 0.183i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.93564 - 0.00612473i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93564 - 0.00612473i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 - 0.629i)T \) |
| 41 | \( 1 + (-6.16 - 1.71i)T \) |
| good | 3 | \( 1 + (-0.761 + 1.83i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.838 + 0.427i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (3.70 - 2.26i)T + (3.17 - 6.23i)T^{2} \) |
| 11 | \( 1 + (-0.486 + 6.18i)T + (-10.8 - 1.72i)T^{2} \) |
| 13 | \( 1 + (2.25 - 0.542i)T + (11.5 - 5.90i)T^{2} \) |
| 17 | \( 1 + (-0.645 - 0.755i)T + (-2.65 + 16.7i)T^{2} \) |
| 19 | \( 1 + (0.0857 - 0.357i)T + (-16.9 - 8.62i)T^{2} \) |
| 23 | \( 1 + (5.47 - 3.97i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.678 + 0.794i)T + (-4.53 - 28.6i)T^{2} \) |
| 31 | \( 1 + (0.466 + 1.43i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.49 + 7.67i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (0.624 + 0.0989i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-7.88 - 4.83i)T + (21.3 + 41.8i)T^{2} \) |
| 53 | \( 1 + (7.73 + 6.60i)T + (8.29 + 52.3i)T^{2} \) |
| 59 | \( 1 + (0.332 + 0.457i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.79 + 0.917i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (0.121 - 0.00957i)T + (66.1 - 10.4i)T^{2} \) |
| 71 | \( 1 + (-9.03 - 0.710i)T + (70.1 + 11.1i)T^{2} \) |
| 73 | \( 1 + (-0.509 + 0.509i)T - 73iT^{2} \) |
| 79 | \( 1 + (8.45 + 3.50i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 14.3iT - 83T^{2} \) |
| 89 | \( 1 + (5.53 + 9.03i)T + (-40.4 + 79.2i)T^{2} \) |
| 97 | \( 1 + (-5.44 + 0.428i)T + (95.8 - 15.1i)T^{2} \) |
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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
−12.95567606723339092836540901647, −12.35950982126257149068704260078, −11.29854959751205068351356915673, −9.556193545420954780806228204629, −8.465708124750050460220950569573, −7.44553189966870163663467601679, −6.18649834807085009925437508648, −5.73526732395791355949894681361, −3.54315789171505124473272500018, −2.39052572042912101906402239000,
2.58253142771184510376479471036, 3.91437647078056765449997006170, 4.66196013702110622208369303848, 6.34683399676062483033727016441, 7.26058229315457605098880535152, 9.499077296125772194167838698449, 10.01158455720840567375388863289, 10.40485952520223538529087905338, 12.17738177017019598044358665890, 12.77015557545493683263479387115
