| L(s) = 1 | + (1.08 − 1.28i)2-s + (−0.796 − 0.605i)3-s + (−0.134 − 0.818i)4-s + (−1.86 − 3.52i)5-s + (−1.64 + 0.361i)6-s + (−3.71 − 0.404i)7-s + (1.68 + 1.01i)8-s + (0.267 + 0.963i)9-s + (−6.54 − 1.44i)10-s + (5.42 + 1.82i)11-s + (−0.388 + 0.733i)12-s + (1.34 − 4.85i)13-s + (−4.56 + 4.32i)14-s + (−0.644 + 3.93i)15-s + (4.71 − 1.58i)16-s + (0.313 − 0.0341i)17-s + ⋯ |
| L(s) = 1 | + (0.770 − 0.906i)2-s + (−0.459 − 0.349i)3-s + (−0.0671 − 0.409i)4-s + (−0.834 − 1.57i)5-s + (−0.670 + 0.147i)6-s + (−1.40 − 0.152i)7-s + (0.596 + 0.358i)8-s + (0.0891 + 0.321i)9-s + (−2.07 − 0.455i)10-s + (1.63 + 0.551i)11-s + (−0.112 + 0.211i)12-s + (0.374 − 1.34i)13-s + (−1.21 + 1.15i)14-s + (−0.166 + 1.01i)15-s + (1.17 − 0.396i)16-s + (0.0761 − 0.00828i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.477681 - 1.15812i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.477681 - 1.15812i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.796 + 0.605i)T \) |
| 59 | \( 1 + (-7.30 - 2.37i)T \) |
| good | 2 | \( 1 + (-1.08 + 1.28i)T + (-0.323 - 1.97i)T^{2} \) |
| 5 | \( 1 + (1.86 + 3.52i)T + (-2.80 + 4.13i)T^{2} \) |
| 7 | \( 1 + (3.71 + 0.404i)T + (6.83 + 1.50i)T^{2} \) |
| 11 | \( 1 + (-5.42 - 1.82i)T + (8.75 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-1.34 + 4.85i)T + (-11.1 - 6.70i)T^{2} \) |
| 17 | \( 1 + (-0.313 + 0.0341i)T + (16.6 - 3.65i)T^{2} \) |
| 19 | \( 1 + (0.0262 - 0.483i)T + (-18.8 - 2.05i)T^{2} \) |
| 23 | \( 1 + (-2.19 + 1.01i)T + (14.8 - 17.5i)T^{2} \) |
| 29 | \( 1 + (0.0575 + 0.0677i)T + (-4.69 + 28.6i)T^{2} \) |
| 31 | \( 1 + (-0.229 - 4.23i)T + (-30.8 + 3.35i)T^{2} \) |
| 37 | \( 1 + (2.38 - 1.43i)T + (17.3 - 32.6i)T^{2} \) |
| 41 | \( 1 + (2.90 + 1.34i)T + (26.5 + 31.2i)T^{2} \) |
| 43 | \( 1 + (-4.42 + 1.49i)T + (34.2 - 26.0i)T^{2} \) |
| 47 | \( 1 + (1.44 - 2.72i)T + (-26.3 - 38.9i)T^{2} \) |
| 53 | \( 1 + (-7.10 + 1.56i)T + (48.1 - 22.2i)T^{2} \) |
| 61 | \( 1 + (5.67 - 6.67i)T + (-9.86 - 60.1i)T^{2} \) |
| 67 | \( 1 + (7.03 + 4.23i)T + (31.3 + 59.1i)T^{2} \) |
| 71 | \( 1 + (-5.48 + 10.3i)T + (-39.8 - 58.7i)T^{2} \) |
| 73 | \( 1 + (4.48 - 4.25i)T + (3.95 - 72.8i)T^{2} \) |
| 79 | \( 1 + (5.92 - 4.50i)T + (21.1 - 76.1i)T^{2} \) |
| 83 | \( 1 + (-5.93 - 14.8i)T + (-60.2 + 57.0i)T^{2} \) |
| 89 | \( 1 + (-8.27 - 9.73i)T + (-14.3 + 87.8i)T^{2} \) |
| 97 | \( 1 + (4.20 + 3.98i)T + (5.25 + 96.8i)T^{2} \) |
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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
−12.30665725956519024863908968312, −11.92742511550485944471441560271, −10.64656530823801678008290010586, −9.420617380067657363908767300372, −8.305843624727473501208450094689, −7.00127429593526948183316443227, −5.53768643405208293340167843537, −4.32730131475232931551919157355, −3.41982655217137074459108438110, −1.09114968819912772078076449822,
3.46524061945891481361983484236, 4.11398116570834590503389600478, 6.10614323565301876862147406473, 6.56067011288150997715194319220, 7.21073999600596691360488858011, 9.102520205561482114690704208917, 10.18089593521313286829188148193, 11.27718227113240436593271608949, 11.94340735903851752142749804641, 13.37566157228420204515916588187
