L(s) = 1 | + 425. i·2-s + (1.21e4 + 1.54e4i)3-s + 8.11e4·4-s + 7.87e5i·5-s + (−6.58e6 + 5.17e6i)6-s − 3.80e7·7-s + 1.46e8i·8-s + (−9.11e7 + 3.76e8i)9-s − 3.35e8·10-s − 3.08e9i·11-s + (9.87e8 + 1.25e9i)12-s + 1.20e10·13-s − 1.61e10i·14-s + (−1.21e10 + 9.58e9i)15-s − 4.08e10·16-s + 1.44e11i·17-s + ⋯ |
L(s) = 1 | + 0.830i·2-s + (0.618 + 0.785i)3-s + 0.309·4-s + 0.403i·5-s + (−0.653 + 0.513i)6-s − 0.943·7-s + 1.08i·8-s + (−0.235 + 0.971i)9-s − 0.335·10-s − 1.30i·11-s + (0.191 + 0.243i)12-s + 1.13·13-s − 0.783i·14-s + (−0.316 + 0.249i)15-s − 0.594·16-s + 1.21i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{19}{2})\) |
\(\approx\) |
\(0.877817 + 1.80777i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.877817 + 1.80777i\) |
\(L(10)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.21e4 - 1.54e4i)T \) |
good | 2 | \( 1 - 425. iT - 2.62e5T^{2} \) |
| 5 | \( 1 - 7.87e5iT - 3.81e12T^{2} \) |
| 7 | \( 1 + 3.80e7T + 1.62e15T^{2} \) |
| 11 | \( 1 + 3.08e9iT - 5.55e18T^{2} \) |
| 13 | \( 1 - 1.20e10T + 1.12e20T^{2} \) |
| 17 | \( 1 - 1.44e11iT - 1.40e22T^{2} \) |
| 19 | \( 1 - 3.34e11T + 1.04e23T^{2} \) |
| 23 | \( 1 + 1.92e12iT - 3.24e24T^{2} \) |
| 29 | \( 1 + 1.09e13iT - 2.10e26T^{2} \) |
| 31 | \( 1 - 2.95e13T + 6.99e26T^{2} \) |
| 37 | \( 1 + 1.11e14T + 1.68e28T^{2} \) |
| 41 | \( 1 + 3.22e14iT - 1.07e29T^{2} \) |
| 43 | \( 1 + 4.11e13T + 2.52e29T^{2} \) |
| 47 | \( 1 - 8.18e14iT - 1.25e30T^{2} \) |
| 53 | \( 1 + 7.04e14iT - 1.08e31T^{2} \) |
| 59 | \( 1 + 3.09e15iT - 7.50e31T^{2} \) |
| 61 | \( 1 + 1.15e16T + 1.36e32T^{2} \) |
| 67 | \( 1 + 5.60e15T + 7.40e32T^{2} \) |
| 71 | \( 1 + 5.38e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 - 1.65e16T + 3.46e33T^{2} \) |
| 79 | \( 1 + 1.66e16T + 1.43e34T^{2} \) |
| 83 | \( 1 - 4.84e16iT - 3.49e34T^{2} \) |
| 89 | \( 1 + 5.03e16iT - 1.22e35T^{2} \) |
| 97 | \( 1 + 1.01e18T + 5.77e35T^{2} \) |
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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
−22.54736376904789266222939608065, −20.85529623403621763430530730844, −19.18100408014114769722870300107, −16.53184342372694803095449548650, −15.55867620595009917719742416165, −13.91955782710433412103737814269, −10.75062418148835970374885015041, −8.459791233845743354907479843072, −6.15854778481231968025290218798, −3.18651791608639209177268081610,
1.24154417391559903032098438629, 3.10475372478871522884758767004, 6.97578087424956301278800764643, 9.539596258056412087260220552730, 12.01801297980422112153158910080, 13.31277110908984203301800330263, 15.79915870126307033203670850997, 18.34715581113816531011922290515, 19.86047568868096861048996241819, 20.68733474552617662851320371422