L(s) = 1 | + (−7.35e4 + 7.35e4i)2-s + (−4.29e6 − 4.29e6i)3-s − 6.51e9i·4-s + (−1.18e11 + 9.62e10i)5-s + 6.31e11·6-s + (1.36e13 − 1.36e13i)7-s + (1.63e14 + 1.63e14i)8-s − 1.81e15i·9-s + (1.62e15 − 1.57e16i)10-s − 1.07e16·11-s + (−2.79e16 + 2.79e16i)12-s + (5.33e17 + 5.33e17i)13-s + 2.00e18i·14-s + (9.21e17 + 9.48e16i)15-s + 3.95e18·16-s + (1.80e19 − 1.80e19i)17-s + ⋯ |
L(s) = 1 | + (−1.12 + 1.12i)2-s + (−0.0997 − 0.0997i)3-s − 1.51i·4-s + (−0.775 + 0.630i)5-s + 0.223·6-s + (0.410 − 0.410i)7-s + (0.580 + 0.580i)8-s − 0.980i·9-s + (0.162 − 1.57i)10-s − 0.234·11-s + (−0.151 + 0.151i)12-s + (0.801 + 0.801i)13-s + 0.921i·14-s + (0.140 + 0.0144i)15-s + 0.214·16-s + (0.370 − 0.370i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(0.4270473993\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4270473993\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.18e11 - 9.62e10i)T \) |
good | 2 | \( 1 + (7.35e4 - 7.35e4i)T - 4.29e9iT^{2} \) |
| 3 | \( 1 + (4.29e6 + 4.29e6i)T + 1.85e15iT^{2} \) |
| 7 | \( 1 + (-1.36e13 + 1.36e13i)T - 1.10e27iT^{2} \) |
| 11 | \( 1 + 1.07e16T + 2.11e33T^{2} \) |
| 13 | \( 1 + (-5.33e17 - 5.33e17i)T + 4.42e35iT^{2} \) |
| 17 | \( 1 + (-1.80e19 + 1.80e19i)T - 2.36e39iT^{2} \) |
| 19 | \( 1 - 4.66e20iT - 8.31e40T^{2} \) |
| 23 | \( 1 + (-7.85e20 - 7.85e20i)T + 3.76e43iT^{2} \) |
| 29 | \( 1 + 2.31e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 - 3.24e23T + 5.29e47T^{2} \) |
| 37 | \( 1 + (-4.94e24 + 4.94e24i)T - 1.52e50iT^{2} \) |
| 41 | \( 1 + 1.24e26T + 4.06e51T^{2} \) |
| 43 | \( 1 + (1.83e25 + 1.83e25i)T + 1.86e52iT^{2} \) |
| 47 | \( 1 + (-3.45e26 + 3.45e26i)T - 3.21e53iT^{2} \) |
| 53 | \( 1 + (3.32e27 + 3.32e27i)T + 1.50e55iT^{2} \) |
| 59 | \( 1 - 3.29e28iT - 4.64e56T^{2} \) |
| 61 | \( 1 - 1.89e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + (1.76e29 - 1.76e29i)T - 2.71e58iT^{2} \) |
| 71 | \( 1 + 1.94e29T + 1.73e59T^{2} \) |
| 73 | \( 1 + (-3.08e29 - 3.08e29i)T + 4.22e59iT^{2} \) |
| 79 | \( 1 - 3.31e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 + (6.01e30 + 6.01e30i)T + 2.57e61iT^{2} \) |
| 89 | \( 1 - 2.42e31iT - 2.40e62T^{2} \) |
| 97 | \( 1 + (-2.36e31 + 2.36e31i)T - 3.77e63iT^{2} \) |
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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
−16.74930485130022040376998593662, −15.51426368255916133219630808700, −14.35822148897825025412509558572, −11.76349414516633949944355053923, −10.08964868450127659113507706264, −8.451771728370416737060363576010, −7.29497163737103243187658190114, −6.14501339695589436046240725445, −3.75394884008639128035696217205, −1.11173939324006904646133475240,
0.24008603235580834813120836065, 1.50687540882365609001345078377, 3.06771567480535353152258221374, 4.99608211693408588750416435581, 7.918016684838769778199612547953, 8.804087732987192936499727516869, 10.58803679019920168669879405685, 11.55840951845712066598571142027, 12.99256594414079634559286338335, 15.56265850723579025246015336905