L(s) = 1 | + (−1.11 − 0.197i)2-s + (−0.522 − 2.96i)3-s + (−0.669 − 0.243i)4-s + (1.89 + 2.26i)5-s + 3.41i·6-s + (1.25 − 1.05i)7-s + (2.66 + 1.53i)8-s + (−5.67 + 2.06i)9-s + (−1.67 − 2.90i)10-s + (−0.588 + 1.01i)11-s + (−0.371 + 2.10i)12-s + (0.543 − 1.49i)13-s + (−1.61 + 0.929i)14-s + (5.71 − 6.80i)15-s + (−1.58 − 1.32i)16-s + (1.29 + 3.56i)17-s + ⋯ |
L(s) = 1 | + (−0.790 − 0.139i)2-s + (−0.301 − 1.70i)3-s + (−0.334 − 0.121i)4-s + (0.849 + 1.01i)5-s + 1.39i·6-s + (0.474 − 0.398i)7-s + (0.942 + 0.544i)8-s + (−1.89 + 0.688i)9-s + (−0.530 − 0.918i)10-s + (−0.177 + 0.307i)11-s + (−0.107 + 0.608i)12-s + (0.150 − 0.413i)13-s + (−0.430 + 0.248i)14-s + (1.47 − 1.75i)15-s + (−0.396 − 0.332i)16-s + (0.314 + 0.863i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.408464 - 0.303968i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.408464 - 0.303968i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-3.80 + 4.74i)T \) |
good | 2 | \( 1 + (1.11 + 0.197i)T + (1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (0.522 + 2.96i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-1.89 - 2.26i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.25 + 1.05i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.588 - 1.01i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.543 + 1.49i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.29 - 3.56i)T + (-13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (1.80 - 0.318i)T + (17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (2.76 - 1.59i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.99 + 2.30i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.07iT - 31T^{2} \) |
| 41 | \( 1 + (0.914 + 0.332i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 - 0.234iT - 43T^{2} \) |
| 47 | \( 1 + (4.78 + 8.28i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.70 - 3.94i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.638 + 0.760i)T + (-10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.95 + 5.37i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-12.0 + 10.1i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.34 - 7.61i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + (7.34 + 8.75i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (6.50 - 2.36i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (4.87 - 5.80i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-11.3 + 6.57i)T + (48.5 - 84.0i)T^{2} \) |
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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
−17.06691569206785068662751078410, −14.54954721604249445605286621200, −13.77446687854350858469045762112, −12.77531747610980739228690670112, −11.12532849394766990948439867255, −10.13929881773372788217912945518, −8.280796186301163152982255713002, −7.21872384596564351995844569676, −5.83479221692238965917945940821, −1.84340815313137188820669159024,
4.40878350334317995904115642845, 5.50021735919517053881495824705, 8.435018560849455313499805098590, 9.312637861832937073774607899142, 10.01930011219452885474430880982, 11.43969227866847611559732169355, 13.23241947174205715569662751501, 14.57898097052653729211769784207, 16.06475059784105628842081018800, 16.65974375587018847449908668076