L(s) = 1 | + (0.295 − 0.754i)2-s + (−0.878 + 1.49i)3-s + (0.985 + 0.913i)4-s + (2.57 − 1.23i)5-s + (0.865 + 1.10i)6-s + (0.114 − 2.64i)7-s + (2.44 − 1.17i)8-s + (−1.45 − 2.62i)9-s + (−0.172 − 2.30i)10-s + (0.206 + 0.259i)11-s + (−2.22 + 0.667i)12-s + (0.950 − 2.42i)13-s + (−1.95 − 0.868i)14-s + (−0.410 + 4.92i)15-s + (0.0368 + 0.491i)16-s + (−3.33 + 3.09i)17-s + ⋯ |
L(s) = 1 | + (0.209 − 0.533i)2-s + (−0.507 + 0.861i)3-s + (0.492 + 0.456i)4-s + (1.14 − 0.553i)5-s + (0.353 + 0.450i)6-s + (0.0432 − 0.999i)7-s + (0.862 − 0.415i)8-s + (−0.485 − 0.874i)9-s + (−0.0546 − 0.729i)10-s + (0.0623 + 0.0782i)11-s + (−0.643 + 0.192i)12-s + (0.263 − 0.671i)13-s + (−0.523 − 0.232i)14-s + (−0.106 + 1.27i)15-s + (0.00921 + 0.122i)16-s + (−0.808 + 0.750i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.81207 - 0.342877i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.81207 - 0.342877i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.878 - 1.49i)T \) |
| 7 | \( 1 + (-0.114 + 2.64i)T \) |
good | 2 | \( 1 + (-0.295 + 0.754i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.57 + 1.23i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-0.206 - 0.259i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.950 + 2.42i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (3.33 - 3.09i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.60 + 4.51i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.899 - 3.93i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-1.02 - 0.315i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (3.59 - 6.23i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.91 - 2.75i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (9.68 + 6.60i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (-1.50 + 1.02i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (4.30 - 10.9i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-2.07 + 0.641i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (4.52 - 3.08i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-8.35 + 7.74i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (7.21 - 12.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.790 - 3.46i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (10.6 + 1.60i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-0.971 - 1.68i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.574 + 1.46i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (2.36 + 6.03i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-3.72 + 6.45i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.96543718668313890899768645845, −10.34240257191311787472890826988, −9.598784942617956047040345947884, −8.571713798871557825138706092329, −7.22892450183213332213936037197, −6.22972297299722313451186446418, −5.13444077171120160327653255937, −4.16890200551048815356722675947, −3.07254756131099423931641105627, −1.39829677150683676807453817241,
1.78479265977573991241636541068, 2.51305325253212691647455362069, 4.92929030091608790216441526149, 5.92706365588227219768259023069, 6.25396624275319198350684052289, 7.11107913516897022195979046685, 8.223933484172963562252943613437, 9.450116062660930292780176692851, 10.34744383942865767729202855837, 11.37346509393026609948245080226