| 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
−12.24483100992952493930355267804, −11.38831756263204112571078561586, −10.50525308515399416408938093354, −10.24125651001901815367193037617, −7.29440631255602458532202405860, −6.73992576150199718220455546813, −5.79288005629026490746558438360, −4.76753197091191253456747899505, −3.62999677654659908294660014119, −2.20406651588791962160571815918,
2.45434032433497647728319741095, 4.25431229072396664405042670515, 5.08024059504231120026147540622, 5.75534561883410059300967365721, 6.72078353926893981956533475746, 7.890257788001101733889393790118, 9.557581506116460191440291876952, 11.02758148319710511935785810524, 11.73585214907605246199573687656, 12.60501375969294484456518707422
