| L(s) = 1 | + (21.4 − 37.2i)2-s + (45.7 + 79.2i)3-s + (−667. − 1.15e3i)4-s + (931. + 1.61e3i)5-s + 3.93e3·6-s + (−4.98e3 − 8.63e3i)7-s − 3.53e4·8-s + (5.65e3 − 9.80e3i)9-s + 8.00e4·10-s − 6.28e4·11-s + (6.10e4 − 1.05e5i)12-s + (−5.76e4 − 9.99e4i)13-s − 4.28e5·14-s + (−8.52e4 + 1.47e5i)15-s + (−4.17e5 + 7.23e5i)16-s + (−3.58e4 + 6.21e4i)17-s + ⋯ |
| L(s) = 1 | + (0.949 − 1.64i)2-s + (0.325 + 0.564i)3-s + (−1.30 − 2.25i)4-s + (0.666 + 1.15i)5-s + 1.23·6-s + (−0.785 − 1.36i)7-s − 3.05·8-s + (0.287 − 0.497i)9-s + 2.53·10-s − 1.29·11-s + (0.849 − 1.47i)12-s + (−0.560 − 0.970i)13-s − 2.98·14-s + (−0.434 + 0.752i)15-s + (−1.59 + 2.75i)16-s + (−0.104 + 0.180i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.336320 + 2.28565i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.336320 + 2.28565i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-9.59e5 - 1.13e7i)T \) |
| good | 2 | \( 1 + (-21.4 + 37.2i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-45.7 - 79.2i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-931. - 1.61e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (4.98e3 + 8.63e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + 6.28e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (5.76e4 + 9.99e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (3.58e4 - 6.21e4i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (2.44e5 + 4.24e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 - 2.54e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.14e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.15e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + (2.61e6 + 4.52e6i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 - 9.29e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.38e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (-1.14e7 + 1.97e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-5.68e7 + 9.85e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.26e7 + 2.18e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (2.41e7 + 4.18e7i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (4.07e6 + 7.06e6i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 - 4.02e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (1.56e8 + 2.70e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-2.84e8 + 4.92e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (2.39e8 - 4.15e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.13e8T + 7.60e17T^{2} \) |
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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
−13.38023482414343523813316247221, −12.83981091418597004107937387130, −10.76412621237455783162599861413, −10.44071372698580101530713436846, −9.597392573363127077667428530546, −6.73257250747487704802670345246, −4.91943519083899950207607483819, −3.38177697130475785859705512110, −2.72355349361898675959310096420, −0.57554623825940626737792992318,
2.52517614169952380210787050506, 4.85950087789259273119599145833, 5.65151521276186036242904799891, 7.04195286684735025123463537395, 8.403750285185108694697764459317, 9.256477430393222251704426067400, 12.56434963691627166251412610585, 12.79954161414293182945606993557, 13.76208322231115510180348680227, 15.07316027387820021908218373994
