| L(s) = 1 | + (0.234 + 0.481i)2-s + (1.05 − 1.34i)4-s + (0.140 + 2.01i)5-s + (2.77 − 0.0968i)7-s + (1.94 + 0.413i)8-s + (−0.935 + 0.539i)10-s + (3.28 − 0.456i)11-s + (−4.91 + 5.09i)13-s + (0.697 + 1.31i)14-s + (−0.571 − 2.29i)16-s + (−0.389 + 3.70i)17-s + (−0.350 + 1.64i)19-s + (2.86 + 1.93i)20-s + (0.991 + 1.47i)22-s + (1.79 − 4.94i)23-s + ⋯ |
| L(s) = 1 | + (0.166 + 0.340i)2-s + (0.527 − 0.674i)4-s + (0.0628 + 0.899i)5-s + (1.04 − 0.0365i)7-s + (0.687 + 0.146i)8-s + (−0.295 + 0.170i)10-s + (0.990 − 0.137i)11-s + (−1.36 + 1.41i)13-s + (0.186 + 0.350i)14-s + (−0.142 − 0.572i)16-s + (−0.0943 + 0.898i)17-s + (−0.0804 + 0.378i)19-s + (0.640 + 0.431i)20-s + (0.211 + 0.314i)22-s + (0.375 − 1.03i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.11650 + 0.826396i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.11650 + 0.826396i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-3.28 + 0.456i)T \) |
| good | 2 | \( 1 + (-0.234 - 0.481i)T + (-1.23 + 1.57i)T^{2} \) |
| 5 | \( 1 + (-0.140 - 2.01i)T + (-4.95 + 0.695i)T^{2} \) |
| 7 | \( 1 + (-2.77 + 0.0968i)T + (6.98 - 0.488i)T^{2} \) |
| 13 | \( 1 + (4.91 - 5.09i)T + (-0.453 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.389 - 3.70i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (0.350 - 1.64i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-1.79 + 4.94i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (5.13 + 2.73i)T + (16.2 + 24.0i)T^{2} \) |
| 31 | \( 1 + (-7.22 + 2.07i)T + (26.2 - 16.4i)T^{2} \) |
| 37 | \( 1 + (4.00 - 0.851i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (-2.17 + 1.15i)T + (22.9 - 33.9i)T^{2} \) |
| 43 | \( 1 + (-0.620 + 0.739i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (7.38 - 5.77i)T + (11.3 - 45.6i)T^{2} \) |
| 53 | \( 1 + (-2.90 - 4.00i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-8.00 + 3.23i)T + (42.4 - 40.9i)T^{2} \) |
| 61 | \( 1 + (0.503 - 1.75i)T + (-51.7 - 32.3i)T^{2} \) |
| 67 | \( 1 + (0.347 + 1.97i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-7.93 - 0.834i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (10.1 - 9.13i)T + (7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-14.2 + 6.96i)T + (48.6 - 62.2i)T^{2} \) |
| 83 | \( 1 + (-4.39 + 4.23i)T + (2.89 - 82.9i)T^{2} \) |
| 89 | \( 1 + (9.25 + 5.34i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.52 + 0.176i)T + (96.0 + 13.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.28102267920527586158668491858, −9.542775488381331743260650129263, −8.436813317823483833259344928522, −7.40977986962725204612631516994, −6.72165536650732110232769815700, −6.14447007034403703184304549547, −4.89251496367971456097961599223, −4.16032629651490507814791768833, −2.47029861290913495204379057158, −1.60337670727985854060567303746,
1.19283210819955449825537839956, 2.42226996737326103248176164384, 3.57041420988184494520801684660, 4.85114321909468895388582681338, 5.19495805423275689632128565425, 6.81810448536986686586515697255, 7.56522323073470064453085568586, 8.269839161235956996483333741992, 9.144550231576652045886525360632, 10.04950478825566053445955957797
