| L(s) = 1 | + (−5.33 + 3.08i)2-s + (3.71 − 6.43i)3-s + (2.97 − 5.15i)4-s + (−14.5 − 8.39i)5-s + 45.8i·6-s + (41.9 − 72.6i)7-s − 160. i·8-s + (93.8 + 162. i)9-s + 103.·10-s + 280.·11-s + (−22.1 − 38.3i)12-s + (907. + 523. i)13-s + 517. i·14-s + (−108. + 62.4i)15-s + (589. + 1.02e3i)16-s + (1.59e3 − 922. i)17-s + ⋯ |
| L(s) = 1 | + (−0.943 + 0.544i)2-s + (0.238 − 0.413i)3-s + (0.0929 − 0.161i)4-s + (−0.260 − 0.150i)5-s + 0.519i·6-s + (0.323 − 0.560i)7-s − 0.886i·8-s + (0.386 + 0.669i)9-s + 0.327·10-s + 0.699·11-s + (−0.0443 − 0.0767i)12-s + (1.48 + 0.859i)13-s + 0.705i·14-s + (−0.124 + 0.0716i)15-s + (0.575 + 0.997i)16-s + (1.34 − 0.773i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.07869 + 0.120939i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07869 + 0.120939i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-3.64e3 - 7.48e3i)T \) |
| good | 2 | \( 1 + (5.33 - 3.08i)T + (16 - 27.7i)T^{2} \) |
| 3 | \( 1 + (-3.71 + 6.43i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (14.5 + 8.39i)T + (1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-41.9 + 72.6i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 - 280.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (-907. - 523. i)T + (1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-1.59e3 + 922. i)T + (7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (1.22e3 + 707. i)T + (1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 - 502. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.17e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.19e3iT - 2.86e7T^{2} \) |
| 41 | \( 1 + (-1.63e3 + 2.83e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + 5.29e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 8.58e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (7.53e3 + 1.30e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-2.61e4 + 1.50e4i)T + (3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.25e4 - 1.30e4i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (3.42e4 - 5.92e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (1.35e3 - 2.34e3i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 - 1.58e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (2.92e4 + 1.68e4i)T + (1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (3.28e4 + 5.68e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-1.87e4 + 1.08e4i)T + (2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.63e5iT - 8.58e9T^{2} \) |
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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
−15.90296310678831387318075772834, −14.16399088790787408649871705030, −13.20014119936210977491855982479, −11.65638837920152979017436263545, −10.13169262070979109366609775156, −8.714914640718737263319093238571, −7.77707665890393149674593044581, −6.62161185942375001287915068565, −4.09995240567991524236204331617, −1.15229921000794008244375446438,
1.28515356868964305728463758992, 3.61309345444076186094045810195, 5.86633089069471947288306218101, 8.115135782566867483861480465346, 9.071145269398153796048390280571, 10.24884657343817494990151401643, 11.29863479564701289524549537714, 12.60779740432523810985855711613, 14.48684368650794516619438021453, 15.21434808427227848656135761550
