L(s) = 1 | − 5.04·2-s − 2.60i·3-s + 17.4·4-s − 13.4i·5-s + 13.1i·6-s + 22.2i·7-s − 47.4·8-s + 20.1·9-s + 67.6i·10-s − 33.6i·11-s − 45.4i·12-s − 73.4·13-s − 111. i·14-s − 35.0·15-s + 100.·16-s + ⋯ |
L(s) = 1 | − 1.78·2-s − 0.502i·3-s + 2.17·4-s − 1.20i·5-s + 0.895i·6-s + 1.19i·7-s − 2.09·8-s + 0.747·9-s + 2.13i·10-s − 0.922i·11-s − 1.09i·12-s − 1.56·13-s − 2.13i·14-s − 0.602·15-s + 1.56·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2634556292\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2634556292\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 + 5.04T + 8T^{2} \) |
| 3 | \( 1 + 2.60iT - 27T^{2} \) |
| 5 | \( 1 + 13.4iT - 125T^{2} \) |
| 7 | \( 1 - 22.2iT - 343T^{2} \) |
| 11 | \( 1 + 33.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 73.4T + 2.19e3T^{2} \) |
| 19 | \( 1 - 42.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 59.2iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 21.3iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 42.2iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 265. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 80.0iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 52.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 551.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 508.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 671. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 859.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 147. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 522. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 245. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 293.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 72.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3iT - 9.12e5T^{2} \) |
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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
−10.58883378073023002454749952899, −9.506305800928353303372870684812, −9.033069290574245006846262901893, −8.148584240299249132058259552865, −7.39514794915099743538908174312, −6.18349571813790401128563941882, −4.95012504601617727351514276912, −2.56719858361110767948959387913, −1.41620354061914891077477217974, −0.18301736546250647593749867043,
1.62729445856647679490918227662, 3.07608566709929112600646700158, 4.64688531951277612130682933365, 6.78897954437709134282179284713, 7.18282112102800051259853147945, 7.84332381862671788018986000270, 9.542406613777555183210130902133, 9.940090457561257145069457295641, 10.46989265285318122259010946163, 11.29170748411136443347638760172