| L(s) = 1 | + (6.36 + 4.62i)2-s + (9.23 + 28.4i)4-s + (59.3 − 43.1i)5-s + (−51.6 − 158. i)7-s + (5.17 − 15.9i)8-s + 576.·10-s + (−3.42 − 401. i)11-s + (−139. − 101. i)13-s + (405. − 1.24e3i)14-s + (879. − 639. i)16-s + (−365. + 265. i)17-s + (−764. + 2.35e3i)19-s + (1.77e3 + 1.28e3i)20-s + (1.83e3 − 2.56e3i)22-s + 2.58e3·23-s + ⋯ |
| L(s) = 1 | + (1.12 + 0.817i)2-s + (0.288 + 0.887i)4-s + (1.06 − 0.771i)5-s + (−0.398 − 1.22i)7-s + (0.0285 − 0.0879i)8-s + 1.82·10-s + (−0.00853 − 0.999i)11-s + (−0.228 − 0.166i)13-s + (0.553 − 1.70i)14-s + (0.859 − 0.624i)16-s + (−0.306 + 0.222i)17-s + (−0.485 + 1.49i)19-s + (0.990 + 0.719i)20-s + (0.807 − 1.13i)22-s + 1.02·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.63816 - 0.516136i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.63816 - 0.516136i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (3.42 + 401. i)T \) |
| good | 2 | \( 1 + (-6.36 - 4.62i)T + (9.88 + 30.4i)T^{2} \) |
| 5 | \( 1 + (-59.3 + 43.1i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (51.6 + 158. i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (139. + 101. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (365. - 265. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (764. - 2.35e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 - 2.58e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-709. - 2.18e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-3.78e3 - 2.74e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-951. - 2.92e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (2.21e3 - 6.81e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 398.T + 1.47e8T^{2} \) |
| 47 | \( 1 + (415. - 1.27e3i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-8.16e3 - 5.93e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.22e4 + 3.77e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.36e4 + 9.89e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + 2.89e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-4.91e4 + 3.57e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.10e4 - 3.39e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.01e4 - 7.35e3i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (1.22e4 - 8.91e3i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + 1.01e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.08e5 - 7.87e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−13.25955685750762744384450723544, −12.47340785144588850971503983460, −10.66081807146411671546292535615, −9.694823810995293229536821013187, −8.219613054493334570967223999912, −6.77529545125080577078949145222, −5.88200876006090047536750821297, −4.79688312046572314532558048332, −3.49693700877059087354082226966, −1.07675001977714696671810436599,
2.22215955867311683647018424101, 2.73955219569218093458704790612, 4.63354492411958287893114421170, 5.76367084533979189179971952161, 6.88778717546459433869884396284, 8.961512312251379972640848898540, 9.990849670941103773793607583996, 11.10570802885776175089570433333, 12.10364263384508900968341498510, 13.01265220919914265152414555018
