L(s) = 1 | + (−6.07e4 − 2.46e4i)2-s + (3.07e9 + 2.99e9i)4-s − 2.49e11·5-s − 4.65e13i·7-s + (−1.12e14 − 2.57e14i)8-s + (1.51e16 + 6.15e15i)10-s − 4.19e16i·11-s − 6.87e17·13-s + (−1.14e18 + 2.82e18i)14-s + (4.88e17 + 1.84e19i)16-s − 4.91e19·17-s − 3.14e20i·19-s + (−7.67e20 − 7.47e20i)20-s + (−1.03e21 + 2.54e21i)22-s + 4.04e21i·23-s + ⋯ |
L(s) = 1 | + (−0.926 − 0.376i)2-s + (0.716 + 0.697i)4-s − 1.63·5-s − 1.39i·7-s + (−0.400 − 0.916i)8-s + (1.51 + 0.615i)10-s − 0.912i·11-s − 1.03·13-s + (−0.526 + 1.29i)14-s + (0.0264 + 0.999i)16-s − 1.01·17-s − 1.08i·19-s + (−1.17 − 1.14i)20-s + (−0.343 + 0.845i)22-s + 0.659i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(0.1704205001\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1704205001\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (6.07e4 + 2.46e4i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.49e11T + 2.32e22T^{2} \) |
| 7 | \( 1 + 4.65e13iT - 1.10e27T^{2} \) |
| 11 | \( 1 + 4.19e16iT - 2.11e33T^{2} \) |
| 13 | \( 1 + 6.87e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + 4.91e19T + 2.36e39T^{2} \) |
| 19 | \( 1 + 3.14e20iT - 8.31e40T^{2} \) |
| 23 | \( 1 - 4.04e21iT - 3.76e43T^{2} \) |
| 29 | \( 1 + 1.16e23T + 6.26e46T^{2} \) |
| 31 | \( 1 - 3.53e23iT - 5.29e47T^{2} \) |
| 37 | \( 1 + 7.58e24T + 1.52e50T^{2} \) |
| 41 | \( 1 - 2.78e25T + 4.06e51T^{2} \) |
| 43 | \( 1 - 1.82e26iT - 1.86e52T^{2} \) |
| 47 | \( 1 - 6.77e25iT - 3.21e53T^{2} \) |
| 53 | \( 1 + 1.13e27T + 1.50e55T^{2} \) |
| 59 | \( 1 + 5.33e27iT - 4.64e56T^{2} \) |
| 61 | \( 1 + 5.47e28T + 1.35e57T^{2} \) |
| 67 | \( 1 - 6.20e28iT - 2.71e58T^{2} \) |
| 71 | \( 1 + 4.47e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 - 5.03e29T + 4.22e59T^{2} \) |
| 79 | \( 1 - 1.79e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 - 2.17e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + 1.19e31T + 2.40e62T^{2} \) |
| 97 | \( 1 + 9.02e31T + 3.77e63T^{2} \) |
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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
−10.89325511615618064088926875827, −9.426788072724051012540236267464, −8.251511693991273160192453526090, −7.44252712225267197796763746921, −6.79879128177722656773933257286, −4.58334534758067949366438580757, −3.71671014074861849782962198136, −2.81536888126771900561899423752, −1.17954646777589952225050580920, −0.26255831259268091188501795138,
0.15521213874434498792703923759, 1.83357376570859394558121611654, 2.71288816146177766752763448842, 4.26708644863641033020966265524, 5.40594141578227700132232471368, 6.78301880474345058601925974236, 7.68835762253022105126347484944, 8.522863530334324451360761141897, 9.469649926294791967439315096434, 10.78225244043116266389216375901