| L(s) = 1 | + (1.81 + 0.197i)2-s + (−0.647 + 0.762i)3-s + (1.29 + 0.284i)4-s + (1.77 + 1.35i)5-s + (−1.32 + 1.25i)6-s + (−0.458 + 1.15i)7-s + (−1.17 − 0.394i)8-s + (−0.161 − 0.986i)9-s + (2.95 + 2.79i)10-s + (2.52 − 1.16i)11-s + (−1.05 + 0.799i)12-s + (0.305 − 1.86i)13-s + (−1.05 + 1.99i)14-s + (−2.17 + 0.479i)15-s + (−4.44 − 2.05i)16-s + (−2.22 − 5.59i)17-s + ⋯ |
| L(s) = 1 | + (1.28 + 0.139i)2-s + (−0.373 + 0.440i)3-s + (0.645 + 0.142i)4-s + (0.794 + 0.603i)5-s + (−0.540 + 0.511i)6-s + (−0.173 + 0.434i)7-s + (−0.414 − 0.139i)8-s + (−0.0539 − 0.328i)9-s + (0.933 + 0.884i)10-s + (0.760 − 0.352i)11-s + (−0.303 + 0.230i)12-s + (0.0846 − 0.516i)13-s + (−0.282 + 0.532i)14-s + (−0.562 + 0.123i)15-s + (−1.11 − 0.514i)16-s + (−0.540 − 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.83621 + 0.690206i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83621 + 0.690206i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.647 - 0.762i)T \) |
| 59 | \( 1 + (-1.39 - 7.55i)T \) |
| good | 2 | \( 1 + (-1.81 - 0.197i)T + (1.95 + 0.429i)T^{2} \) |
| 5 | \( 1 + (-1.77 - 1.35i)T + (1.33 + 4.81i)T^{2} \) |
| 7 | \( 1 + (0.458 - 1.15i)T + (-5.08 - 4.81i)T^{2} \) |
| 11 | \( 1 + (-2.52 + 1.16i)T + (7.12 - 8.38i)T^{2} \) |
| 13 | \( 1 + (-0.305 + 1.86i)T + (-12.3 - 4.15i)T^{2} \) |
| 17 | \( 1 + (2.22 + 5.59i)T + (-12.3 + 11.6i)T^{2} \) |
| 19 | \( 1 + (-1.25 - 1.84i)T + (-7.03 + 17.6i)T^{2} \) |
| 23 | \( 1 + (0.254 - 4.69i)T + (-22.8 - 2.48i)T^{2} \) |
| 29 | \( 1 + (7.85 - 0.853i)T + (28.3 - 6.23i)T^{2} \) |
| 31 | \( 1 + (-2.07 + 3.06i)T + (-11.4 - 28.7i)T^{2} \) |
| 37 | \( 1 + (0.409 - 0.137i)T + (29.4 - 22.3i)T^{2} \) |
| 41 | \( 1 + (0.412 + 7.60i)T + (-40.7 + 4.43i)T^{2} \) |
| 43 | \( 1 + (-5.14 - 2.37i)T + (27.8 + 32.7i)T^{2} \) |
| 47 | \( 1 + (3.88 - 2.95i)T + (12.5 - 45.2i)T^{2} \) |
| 53 | \( 1 + (5.08 - 4.81i)T + (2.86 - 52.9i)T^{2} \) |
| 61 | \( 1 + (-13.1 - 1.43i)T + (59.5 + 13.1i)T^{2} \) |
| 67 | \( 1 + (-4.13 - 1.39i)T + (53.3 + 40.5i)T^{2} \) |
| 71 | \( 1 + (7.27 - 5.53i)T + (18.9 - 68.4i)T^{2} \) |
| 73 | \( 1 + (1.95 - 3.68i)T + (-40.9 - 60.4i)T^{2} \) |
| 79 | \( 1 + (0.176 + 0.207i)T + (-12.7 + 77.9i)T^{2} \) |
| 83 | \( 1 + (10.1 - 6.09i)T + (38.8 - 73.3i)T^{2} \) |
| 89 | \( 1 + (-8.71 + 0.947i)T + (86.9 - 19.1i)T^{2} \) |
| 97 | \( 1 + (3.74 + 7.05i)T + (-54.4 + 80.2i)T^{2} \) |
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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
−12.99435555678010067014425918468, −11.88382682946782949467862133709, −11.17042156119575415456104408420, −9.792242062279980155845300973981, −9.098936967214567644102822860065, −7.11426201088589031577854971775, −5.97906229802235746138947320521, −5.44954483052940532013343869951, −4.00301161031207632360232296672, −2.77133987450420849418332331917,
1.90736794355786768926975453672, 3.86531785985194364809782797461, 4.92455565359185208701799127654, 6.05531471449391418129456475963, 6.81693396160967441577451512301, 8.583242411433464160695156984112, 9.608078708378955104407385767112, 10.98017103589757005372213022344, 11.93370607447814125556627201416, 12.92867980023110817870737415679
