| L(s) = 1 | + (14.6 − 10.6i)2-s + (61.5 − 189. i)4-s + (142. + 103. i)5-s + (−426. + 1.31e3i)7-s + (−398. − 1.22e3i)8-s + 3.17e3·10-s + (−827. + 4.33e3i)11-s + (7.18e3 − 5.21e3i)13-s + (7.71e3 + 2.37e4i)14-s + (1.77e3 + 1.29e3i)16-s + (6.42e3 + 4.67e3i)17-s + (1.68e4 + 5.20e4i)19-s + (2.83e4 − 2.05e4i)20-s + (3.40e4 + 7.22e4i)22-s − 6.90e4·23-s + ⋯ |
| L(s) = 1 | + (1.29 − 0.939i)2-s + (0.480 − 1.48i)4-s + (0.508 + 0.369i)5-s + (−0.470 + 1.44i)7-s + (−0.274 − 0.845i)8-s + 1.00·10-s + (−0.187 + 0.982i)11-s + (0.906 − 0.658i)13-s + (0.751 + 2.31i)14-s + (0.108 + 0.0788i)16-s + (0.317 + 0.230i)17-s + (0.565 + 1.73i)19-s + (0.791 − 0.575i)20-s + (0.680 + 1.44i)22-s − 1.18·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.20628 + 0.214516i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.20628 + 0.214516i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (827. - 4.33e3i)T \) |
| good | 2 | \( 1 + (-14.6 + 10.6i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (-142. - 103. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (426. - 1.31e3i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-7.18e3 + 5.21e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-6.42e3 - 4.67e3i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-1.68e4 - 5.20e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + 6.90e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-3.04e4 + 9.36e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-327. + 237. i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (7.96e3 - 2.45e4i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (8.87e4 + 2.73e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 3.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.64e5 - 8.14e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-1.71e6 + 1.24e6i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (1.67e5 - 5.16e5i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (9.32e4 + 6.77e4i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 + 4.50e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (2.52e5 + 1.83e5i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (3.37e5 - 1.03e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (6.98e4 - 5.07e4i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-5.38e6 - 3.91e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 3.18e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (7.39e6 - 5.37e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−12.29637481121692462001132414878, −12.03128299629550434373918931486, −10.48345945727304438878096742443, −9.779321149082134775304365714970, −8.141846015288100895884769278286, −6.03917116088862211723937809284, −5.62268254407958337431415181102, −3.94890544418101374778524598856, −2.72004812910089990635571284181, −1.77565755006443550031352525696,
0.875315556495923127527944027388, 3.28605547839922819238595705589, 4.31200803688416631310863469796, 5.56974908671395625128716850667, 6.59911053581541900713717875269, 7.51715618976444014187655543249, 9.034064042436640539327518648723, 10.45352334222482387313124632946, 11.69656956522808882671806004218, 13.21660518750383573816500435267
