| L(s) = 1 | + (5.80 − 7.98i)2-s + (18.8 − 6.11i)3-s + (−20.2 − 62.2i)4-s + (−7.83 + 55.3i)5-s + (60.3 − 185. i)6-s + 206. i·7-s + (−314. − 102. i)8-s + (119. − 86.9i)9-s + (396. + 383. i)10-s + (−356. − 258. i)11-s + (−760. − 1.04e3i)12-s + (−361. − 497. i)13-s + (1.65e3 + 1.19e3i)14-s + (190. + 1.08e3i)15-s + (−943. + 685. i)16-s + (979. + 318. i)17-s + ⋯ |
| L(s) = 1 | + (1.02 − 1.41i)2-s + (1.20 − 0.391i)3-s + (−0.632 − 1.94i)4-s + (−0.140 + 0.990i)5-s + (0.684 − 2.10i)6-s + 1.59i·7-s + (−1.73 − 0.563i)8-s + (0.492 − 0.358i)9-s + (1.25 + 1.21i)10-s + (−0.888 − 0.645i)11-s + (−1.52 − 2.09i)12-s + (−0.592 − 0.815i)13-s + (2.25 + 1.63i)14-s + (0.219 + 1.24i)15-s + (−0.921 + 0.669i)16-s + (0.821 + 0.267i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0926 + 0.995i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0926 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.98932 - 2.18294i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98932 - 2.18294i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (7.83 - 55.3i)T \) |
| good | 2 | \( 1 + (-5.80 + 7.98i)T + (-9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (-18.8 + 6.11i)T + (196. - 142. i)T^{2} \) |
| 7 | \( 1 - 206. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (356. + 258. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (361. + 497. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-979. - 318. i)T + (1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-230. + 708. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-665. + 915. i)T + (-1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-821. - 2.52e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-2.43e3 + 7.49e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-4.58e3 - 6.31e3i)T + (-2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (3.87e3 - 2.81e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 4.73e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.00e3 + 326. i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-232. + 75.6i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.56e4 - 1.13e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-2.47e4 - 1.80e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-2.71e4 - 8.82e3i)T + (1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-1.60e4 - 4.93e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-4.19e4 + 5.77e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.11e4 - 3.44e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (4.04e4 + 1.31e4i)T + (3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (6.47e4 + 4.70e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (1.33e5 - 4.32e4i)T + (6.94e9 - 5.04e9i)T^{2} \) |
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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
−15.24940344926047482538415161605, −14.69726333175844305974263449094, −13.52012800422341791722201884081, −12.50026451824480119831690669736, −11.25750003058880096827091369666, −9.838556306868803792715017230207, −8.131925451683932326986059264249, −5.57720321422522531012903172895, −3.02279997558372624938526615149, −2.52157527866306526532701858198,
3.80559384740876563265399954854, 4.90790065785434261794274304236, 7.27748473185136714517780939196, 8.144003580218097359810003162998, 9.769581917283454984654669192697, 12.51099455215420617188405234523, 13.70715729837922256984252181003, 14.25297636498796204852589145072, 15.51787381694560869718590995191, 16.46979748591805545923003066877
