| L(s) = 1 | + (2.68 + 0.570i)2-s + (−1.72 + 0.150i)3-s + (5.05 + 2.24i)4-s + (0.420 − 2.19i)5-s + (−4.71 − 0.580i)6-s + (−0.931 + 1.61i)7-s + (7.83 + 5.69i)8-s + (2.95 − 0.519i)9-s + (2.38 − 5.65i)10-s + (−2.70 − 0.574i)11-s + (−9.05 − 3.11i)12-s + (−1.65 + 0.352i)13-s + (−3.41 + 3.79i)14-s + (−0.394 + 3.85i)15-s + (10.3 + 11.5i)16-s + (−5.11 − 3.71i)17-s + ⋯ |
| L(s) = 1 | + (1.89 + 0.403i)2-s + (−0.996 + 0.0869i)3-s + (2.52 + 1.12i)4-s + (0.188 − 0.982i)5-s + (−1.92 − 0.236i)6-s + (−0.351 + 0.609i)7-s + (2.77 + 2.01i)8-s + (0.984 − 0.173i)9-s + (0.753 − 1.78i)10-s + (−0.814 − 0.173i)11-s + (−2.61 − 0.900i)12-s + (−0.459 + 0.0976i)13-s + (−0.913 + 1.01i)14-s + (−0.101 + 0.994i)15-s + (2.59 + 2.88i)16-s + (−1.24 − 0.901i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.50916 + 0.669836i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.50916 + 0.669836i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 - 0.150i)T \) |
| 5 | \( 1 + (-0.420 + 2.19i)T \) |
| good | 2 | \( 1 + (-2.68 - 0.570i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (0.931 - 1.61i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.70 + 0.574i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.65 - 0.352i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (5.11 + 3.71i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.23 - 0.896i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.327 + 0.363i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.189 + 1.79i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.285 + 2.71i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-1.30 + 4.01i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.71 + 0.790i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (5.30 - 9.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.670 - 6.38i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-4.80 + 3.49i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-9.93 + 2.11i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (7.13 + 1.51i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.914 + 8.69i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-2.74 + 1.99i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.80 - 14.7i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.34 - 12.7i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (0.803 - 0.357i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-1.51 - 4.65i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.396 - 3.77i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.60501375969294484456518707422, −11.73585214907605246199573687656, −11.02758148319710511935785810524, −9.557581506116460191440291876952, −7.890257788001101733889393790118, −6.72078353926893981956533475746, −5.75534561883410059300967365721, −5.08024059504231120026147540622, −4.25431229072396664405042670515, −2.45434032433497647728319741095,
2.20406651588791962160571815918, 3.62999677654659908294660014119, 4.76753197091191253456747899505, 5.79288005629026490746558438360, 6.73992576150199718220455546813, 7.29440631255602458532202405860, 10.24125651001901815367193037617, 10.50525308515399416408938093354, 11.38831756263204112571078561586, 12.24483100992952493930355267804
