| L(s) = 1 | + (−1.90 − 1.76i)2-s + (0.353 + 4.72i)4-s + (1.29 + 3.29i)5-s + (−2.54 − 0.706i)7-s + (4.42 − 5.54i)8-s + (3.35 − 8.55i)10-s + (−0.000902 + 0.000278i)11-s + (0.557 − 2.44i)13-s + (3.60 + 5.84i)14-s + (−8.84 + 1.33i)16-s + (−1.28 − 0.875i)17-s + (−4.05 + 7.02i)19-s + (−15.1 + 7.27i)20-s + (0.00220 + 0.00106i)22-s + (−0.736 + 0.502i)23-s + ⋯ |
| L(s) = 1 | + (−1.34 − 1.24i)2-s + (0.176 + 2.36i)4-s + (0.578 + 1.47i)5-s + (−0.963 − 0.267i)7-s + (1.56 − 1.96i)8-s + (1.06 − 2.70i)10-s + (−0.000272 + 8.39e−5i)11-s + (0.154 − 0.676i)13-s + (0.962 + 1.56i)14-s + (−2.21 + 0.333i)16-s + (−0.311 − 0.212i)17-s + (−0.930 + 1.61i)19-s + (−3.37 + 1.62i)20-s + (0.000471 + 0.000226i)22-s + (−0.153 + 0.104i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0600 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0600 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.247754 + 0.233288i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.247754 + 0.233288i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.54 + 0.706i)T \) |
| good | 2 | \( 1 + (1.90 + 1.76i)T + (0.149 + 1.99i)T^{2} \) |
| 5 | \( 1 + (-1.29 - 3.29i)T + (-3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (0.000902 - 0.000278i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.557 + 2.44i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (1.28 + 0.875i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (4.05 - 7.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.736 - 0.502i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (6.27 - 3.02i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-0.104 - 0.180i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.617 - 8.24i)T + (-36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (1.75 - 2.20i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (6.37 + 7.99i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-0.166 - 0.154i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.614 - 8.19i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (0.205 - 0.523i)T + (-43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (-0.734 + 9.80i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (1.88 + 3.26i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (8.36 + 4.02i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (6.17 - 5.73i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (5.86 - 10.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.21 + 5.32i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-10.3 - 3.18i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−10.91513144309006381074950764868, −10.26973427765573761824367202483, −9.997585826116707999328902466460, −8.944285852883731565593853621425, −7.86627216463839060686483367466, −6.95856576705293074034896493397, −6.04242998793141281211864162190, −3.67449285300241867913357632797, −3.02395070501005302056261463072, −1.87102468332412035896172440144,
0.31589708437356655380057886719, 1.93979895006735516001277804882, 4.49532527670174874892763299181, 5.59747046219269586909129987628, 6.33982719750119833414516877794, 7.22224522057193771739146530438, 8.502552312444416816901104633267, 9.026384370836189600137676079566, 9.460348252455171633724378064608, 10.40057060242166538932420590452
