| L(s) = 1 | + (−0.845 + 0.186i)2-s + (−0.511 + 1.65i)3-s + (−1.13 + 0.525i)4-s + (−1.97 − 0.547i)5-s + (0.124 − 1.49i)6-s + (0.395 − 0.374i)7-s + (2.24 − 1.70i)8-s + (−2.47 − 1.69i)9-s + (1.76 + 0.0959i)10-s + (−2.26 − 2.66i)11-s + (−0.288 − 2.14i)12-s + (−0.338 − 1.00i)13-s + (−0.264 + 0.390i)14-s + (1.91 − 2.98i)15-s + (0.0419 − 0.0493i)16-s + (−2.83 + 2.99i)17-s + ⋯ |
| L(s) = 1 | + (−0.597 + 0.131i)2-s + (−0.295 + 0.955i)3-s + (−0.567 + 0.262i)4-s + (−0.882 − 0.244i)5-s + (0.0508 − 0.610i)6-s + (0.149 − 0.141i)7-s + (0.792 − 0.602i)8-s + (−0.825 − 0.564i)9-s + (0.559 + 0.0303i)10-s + (−0.682 − 0.803i)11-s + (−0.0832 − 0.619i)12-s + (−0.0937 − 0.278i)13-s + (−0.0707 + 0.104i)14-s + (0.494 − 0.770i)15-s + (0.0104 − 0.0123i)16-s + (−0.688 + 0.726i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00624589 - 0.0133138i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00624589 - 0.0133138i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.511 - 1.65i)T \) |
| 59 | \( 1 + (7.14 - 2.81i)T \) |
| good | 2 | \( 1 + (0.845 - 0.186i)T + (1.81 - 0.839i)T^{2} \) |
| 5 | \( 1 + (1.97 + 0.547i)T + (4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-0.395 + 0.374i)T + (0.378 - 6.98i)T^{2} \) |
| 11 | \( 1 + (2.26 + 2.66i)T + (-1.77 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.338 + 1.00i)T + (-10.3 + 7.86i)T^{2} \) |
| 17 | \( 1 + (2.83 - 2.99i)T + (-0.920 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.103 + 0.259i)T + (-13.7 + 13.0i)T^{2} \) |
| 23 | \( 1 + (4.94 - 0.537i)T + (22.4 - 4.94i)T^{2} \) |
| 29 | \( 1 + (1.72 - 7.83i)T + (-26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (7.50 + 2.99i)T + (22.5 + 21.3i)T^{2} \) |
| 37 | \( 1 + (-0.225 + 0.296i)T + (-9.89 - 35.6i)T^{2} \) |
| 41 | \( 1 + (-0.267 + 2.46i)T + (-40.0 - 8.81i)T^{2} \) |
| 43 | \( 1 + (-5.32 - 4.52i)T + (6.95 + 42.4i)T^{2} \) |
| 47 | \( 1 + (1.24 + 4.49i)T + (-40.2 + 24.2i)T^{2} \) |
| 53 | \( 1 + (-12.2 + 0.665i)T + (52.6 - 5.73i)T^{2} \) |
| 61 | \( 1 + (0.156 + 0.710i)T + (-55.3 + 25.6i)T^{2} \) |
| 67 | \( 1 + (6.26 + 8.24i)T + (-17.9 + 64.5i)T^{2} \) |
| 71 | \( 1 + (0.609 - 0.169i)T + (60.8 - 36.6i)T^{2} \) |
| 73 | \( 1 + (9.03 + 6.12i)T + (27.0 + 67.8i)T^{2} \) |
| 79 | \( 1 + (-1.97 - 12.0i)T + (-74.8 + 25.2i)T^{2} \) |
| 83 | \( 1 + (5.88 + 11.1i)T + (-46.5 + 68.6i)T^{2} \) |
| 89 | \( 1 + (3.67 + 0.809i)T + (80.7 + 37.3i)T^{2} \) |
| 97 | \( 1 + (0.834 - 0.565i)T + (35.9 - 90.1i)T^{2} \) |
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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
−12.23115636563674823892823973461, −11.03191897090376918133182739343, −10.40097830781109444067202309742, −9.161839971277975792296201516345, −8.424380714636112649753599794298, −7.54273571250265967406423645220, −5.72461964845597022187883897566, −4.45155853367002264420050935704, −3.57425510250051890690870452400, −0.01664389365051062828901763738,
2.13414635910311457341336077170, 4.34776488566742762421248607079, 5.62431672266750880471561726630, 7.22128126956858574929071731034, 7.83759354929247621992682630855, 8.876578133219111254523995173178, 10.09664666749882282988950601300, 11.17845705187768915503255911676, 11.91951357306502483698085809132, 13.00036649079798641232103056865
