| L(s) = 1 | + (−0.0979 + 0.116i)2-s + (−67.1 + 56.3i)3-s + (88.9 + 504. i)4-s + (−720. + 1.97e3i)5-s − 13.3i·6-s + (1.08e4 + 3.93e3i)7-s + (−135. − 77.9i)8-s + (−2.08e3 + 1.18e4i)9-s + (−160. − 277. i)10-s + (1.02e4 − 1.77e4i)11-s + (−3.43e4 − 2.88e4i)12-s + (−4.69e4 + 8.27e3i)13-s + (−1.51e3 + 875. i)14-s + (−6.31e4 − 1.73e5i)15-s + (−2.46e5 + 8.96e4i)16-s + (5.37e5 + 9.47e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.00432 + 0.00515i)2-s + (−0.478 + 0.401i)3-s + (0.173 + 0.984i)4-s + (−0.515 + 1.41i)5-s − 0.00420i·6-s + (1.70 + 0.619i)7-s + (−0.0116 − 0.00673i)8-s + (−0.105 + 0.600i)9-s + (−0.00507 − 0.00878i)10-s + (0.210 − 0.365i)11-s + (−0.478 − 0.401i)12-s + (−0.455 + 0.0803i)13-s + (−0.0105 + 0.00609i)14-s + (−0.321 − 0.884i)15-s + (−0.939 + 0.341i)16-s + (1.56 + 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.141048 + 1.70245i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.141048 + 1.70245i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (6.08e6 + 9.63e6i)T \) |
| good | 2 | \( 1 + (0.0979 - 0.116i)T + (-88.9 - 504. i)T^{2} \) |
| 3 | \( 1 + (67.1 - 56.3i)T + (3.41e3 - 1.93e4i)T^{2} \) |
| 5 | \( 1 + (720. - 1.97e3i)T + (-1.49e6 - 1.25e6i)T^{2} \) |
| 7 | \( 1 + (-1.08e4 - 3.93e3i)T + (3.09e7 + 2.59e7i)T^{2} \) |
| 11 | \( 1 + (-1.02e4 + 1.77e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (4.69e4 - 8.27e3i)T + (9.96e9 - 3.62e9i)T^{2} \) |
| 17 | \( 1 + (-5.37e5 - 9.47e4i)T + (1.11e11 + 4.05e10i)T^{2} \) |
| 19 | \( 1 + (-2.80e5 - 3.33e5i)T + (-5.60e10 + 3.17e11i)T^{2} \) |
| 23 | \( 1 + (1.01e6 - 5.86e5i)T + (9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-3.25e6 - 1.88e6i)T + (7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + 9.14e6iT - 2.64e13T^{2} \) |
| 41 | \( 1 + (-2.50e6 - 1.42e7i)T + (-3.07e14 + 1.11e14i)T^{2} \) |
| 43 | \( 1 + 3.14e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (-3.55e6 - 6.15e6i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-1.00e8 + 3.64e7i)T + (2.52e15 - 2.12e15i)T^{2} \) |
| 59 | \( 1 + (8.57e6 + 2.35e7i)T + (-6.63e15 + 5.56e15i)T^{2} \) |
| 61 | \( 1 + (-1.99e8 + 3.52e7i)T + (1.09e16 - 3.99e15i)T^{2} \) |
| 67 | \( 1 + (2.78e7 + 1.01e7i)T + (2.08e16 + 1.74e16i)T^{2} \) |
| 71 | \( 1 + (1.77e8 - 1.49e8i)T + (7.96e15 - 4.51e16i)T^{2} \) |
| 73 | \( 1 + 1.58e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (-2.66e7 + 7.32e7i)T + (-9.18e16 - 7.70e16i)T^{2} \) |
| 83 | \( 1 + (-3.02e7 + 1.71e8i)T + (-1.75e17 - 6.39e16i)T^{2} \) |
| 89 | \( 1 + (-2.54e8 - 7.00e8i)T + (-2.68e17 + 2.25e17i)T^{2} \) |
| 97 | \( 1 + (4.90e8 - 2.83e8i)T + (3.80e17 - 6.58e17i)T^{2} \) |
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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
−14.87345403070256261224394188934, −14.07571959016517896214850612355, −11.88393022839060688373040788875, −11.52035708446652493326864049029, −10.37945628348759092022586395693, −8.157584256752785753392640284259, −7.49650831026967471765432236654, −5.53407385396058398701743531956, −3.85418792815856950884105132901, −2.31032267876696897412176009673,
0.77874595079107439789238925194, 1.33054997064755666709729075639, 4.57927490161740636980925503181, 5.40459435719901467120351889063, 7.27947209838482786988859041075, 8.604204324308948751967129903560, 10.13911551171252561179239219379, 11.66283559792311691036512679935, 12.18978402259573715700035883399, 13.97514195647843877696639440876
