L(s) = 1 | + (5.60e4 + 3.40e4i)2-s + (1.98e9 + 3.81e9i)4-s + 2.17e11·5-s − 2.81e12i·7-s + (−1.85e13 + 2.80e14i)8-s + (1.21e16 + 7.38e15i)10-s − 6.95e16i·11-s − 7.08e17·13-s + (9.58e16 − 1.57e17i)14-s + (−1.05e19 + 1.51e19i)16-s − 6.55e19·17-s + 1.83e20i·19-s + (4.30e20 + 8.27e20i)20-s + (2.36e21 − 3.89e21i)22-s + 7.92e20i·23-s + ⋯ |
L(s) = 1 | + (0.854 + 0.518i)2-s + (0.461 + 0.887i)4-s + 1.42·5-s − 0.0847i·7-s + (−0.0658 + 0.997i)8-s + (1.21 + 0.738i)10-s − 1.51i·11-s − 1.06·13-s + (0.0440 − 0.0724i)14-s + (−0.574 + 0.818i)16-s − 1.34·17-s + 0.637i·19-s + (0.656 + 1.26i)20-s + (0.785 − 1.29i)22-s + 0.129i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(3.093539524\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.093539524\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-5.60e4 - 3.40e4i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2.17e11T + 2.32e22T^{2} \) |
| 7 | \( 1 + 2.81e12iT - 1.10e27T^{2} \) |
| 11 | \( 1 + 6.95e16iT - 2.11e33T^{2} \) |
| 13 | \( 1 + 7.08e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + 6.55e19T + 2.36e39T^{2} \) |
| 19 | \( 1 - 1.83e20iT - 8.31e40T^{2} \) |
| 23 | \( 1 - 7.92e20iT - 3.76e43T^{2} \) |
| 29 | \( 1 - 1.45e23T + 6.26e46T^{2} \) |
| 31 | \( 1 + 1.34e24iT - 5.29e47T^{2} \) |
| 37 | \( 1 + 2.62e24T + 1.52e50T^{2} \) |
| 41 | \( 1 - 1.80e25T + 4.06e51T^{2} \) |
| 43 | \( 1 + 1.56e26iT - 1.86e52T^{2} \) |
| 47 | \( 1 + 8.64e26iT - 3.21e53T^{2} \) |
| 53 | \( 1 + 5.50e27T + 1.50e55T^{2} \) |
| 59 | \( 1 + 9.48e27iT - 4.64e56T^{2} \) |
| 61 | \( 1 - 3.04e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + 7.40e28iT - 2.71e58T^{2} \) |
| 71 | \( 1 - 2.60e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 - 3.70e29T + 4.22e59T^{2} \) |
| 79 | \( 1 - 2.10e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 + 6.24e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + 1.03e31T + 2.40e62T^{2} \) |
| 97 | \( 1 - 1.05e32T + 3.77e63T^{2} \) |
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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
−10.59971371176641366806379951911, −9.305210378144218436001475613747, −8.197443202524250338627491040069, −6.81022226065982468709512718483, −5.95100789988003054644663016221, −5.23330954193285396517153967204, −3.95455776620220748362187180800, −2.66317029059693065604099384428, −1.96054027110560280212545631220, −0.31762525136755202434344394540,
1.29043756346559338794551891139, 2.17130813241284232934212339650, 2.73604280712646512215774529989, 4.54266785351539910365784209378, 5.03456091062719702060847884063, 6.34578732494045037544764641358, 7.09128930669927182263081433879, 9.209095387062100535055216209599, 9.899953866290359074232358554487, 10.82886049890279753994236055521