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 | \( 1 + (-2.36e31 - 2.36e31i)T + 3.77e63iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.56265850723579025246015336905, −12.99256594414079634559286338335, −11.55840951845712066598571142027, −10.58803679019920168669879405685, −8.804087732987192936499727516869, −7.918016684838769778199612547953, −4.99608211693408588750416435581, −3.06771567480535353152258221374, −1.50687540882365609001345078377, −0.24008603235580834813120836065,
1.11173939324006904646133475240, 3.75394884008639128035696217205, 6.14501339695589436046240725445, 7.29497163737103243187658190114, 8.451771728370416737060363576010, 10.08964868450127659113507706264, 11.76349414516633949944355053923, 14.35822148897825025412509558572, 15.51426368255916133219630808700, 16.74930485130022040376998593662