L(s) = 1 | − 5.31i·2-s − 3·3-s − 20.2·4-s + 13.7i·5-s + 15.9i·6-s + 18.9i·7-s + 64.8i·8-s + 9·9-s + 73.0·10-s − 23.9·11-s + 60.6·12-s − 9.51·13-s + 100.·14-s − 41.2i·15-s + 182.·16-s − 35.2·17-s + ⋯ |
L(s) = 1 | − 1.87i·2-s − 0.577·3-s − 2.52·4-s + 1.22i·5-s + 1.08i·6-s + 1.02i·7-s + 2.86i·8-s + 0.333·9-s + 2.30·10-s − 0.655·11-s + 1.45·12-s − 0.202·13-s + 1.91·14-s − 0.709i·15-s + 2.85·16-s − 0.502·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3425538989\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3425538989\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 157 | \( 1 + (1.87e3 + 605. i)T \) |
good | 2 | \( 1 + 5.31iT - 8T^{2} \) |
| 5 | \( 1 - 13.7iT - 125T^{2} \) |
| 7 | \( 1 - 18.9iT - 343T^{2} \) |
| 11 | \( 1 + 23.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 9.51T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 80.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 271. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 33.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 286.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 249. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 83.6iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 389.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 370. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 8.91iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 46.9iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 318.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 900.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 972. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.15e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 168. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 206.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.49e3iT - 9.12e5T^{2} \) |
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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
−10.54671579483070927386968012391, −9.617240537040664091589605575355, −8.761327183958324120898922622688, −7.48896704719744714671019165401, −6.04559282187192157011190177240, −5.09776668166882610199079410581, −3.85123700346808724366886539456, −2.73160259063009543599109113961, −2.03749929670853597340285389055, −0.15359357851305236728209432910,
0.935142950976527271498543140557, 4.00640553437958781166384753996, 4.87952953416094321703890203709, 5.37732979325015683314895051735, 6.61730898882866945737695956283, 7.26126161391298760818379230756, 8.258664066657334898108956768258, 8.895140315144422701131703089878, 9.929850045174237884569405469212, 10.86129099810112415277454371676