| L(s) = 1 | + (−3.80 + 1.23i)2-s + (−14.7 − 20.3i)3-s + (12.9 − 9.40i)4-s + (−53.1 − 17.3i)5-s + (81.2 + 59.0i)6-s − 248. i·7-s + (−37.6 + 51.7i)8-s + (−119. + 367. i)9-s + (223. + 0.495i)10-s + (87.4 + 269. i)11-s + (−381. − 124. i)12-s + (88.2 + 28.6i)13-s + (307. + 945. i)14-s + (430. + 1.33e3i)15-s + (79.1 − 243. i)16-s + (−833. + 1.14e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (−0.946 − 1.30i)3-s + (0.404 − 0.293i)4-s + (−0.950 − 0.311i)5-s + (0.921 + 0.669i)6-s − 1.91i·7-s + (−0.207 + 0.286i)8-s + (−0.492 + 1.51i)9-s + (0.707 + 0.00156i)10-s + (0.217 + 0.670i)11-s + (−0.765 − 0.248i)12-s + (0.144 + 0.0470i)13-s + (0.418 + 1.28i)14-s + (0.494 + 1.53i)15-s + (0.0772 − 0.237i)16-s + (−0.699 + 0.962i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0989129 + 0.144853i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0989129 + 0.144853i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.80 - 1.23i)T \) |
| 5 | \( 1 + (53.1 + 17.3i)T \) |
| good | 3 | \( 1 + (14.7 + 20.3i)T + (-75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + 248. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-87.4 - 269. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-88.2 - 28.6i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (833. - 1.14e3i)T + (-4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (1.40e3 + 1.01e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-3.75e3 + 1.21e3i)T + (5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-2.94e3 + 2.14e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.44e3 - 1.04e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (7.39e3 + 2.40e3i)T + (5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-935. + 2.88e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 6.09e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (1.09e4 + 1.50e4i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (5.01e3 + 6.90e3i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.33e4 - 4.10e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-4.19e3 - 1.28e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (9.12e3 - 1.25e4i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (2.02e4 - 1.47e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.16e4 - 3.77e3i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-4.80e4 + 3.48e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (5.96e4 - 8.20e4i)T + (-1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-6.19e3 - 1.90e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (7.99e4 + 1.10e5i)T + (-2.65e9 + 8.16e9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.42336430168445550001848318948, −12.56575249357418155080592616006, −11.26122819829576001893842245783, −10.54935334595225855925790626773, −8.405266800894248828164810301944, −7.17008859206980900317048071446, −6.73694779262975695756916491139, −4.46106919379193149988346063334, −1.23586004373570527072055367652, −0.14238038439218516468292644413,
3.13137763445547810139092821623, 4.95059972511666706524900540585, 6.36474108066216814261309115066, 8.505952753982249714961740528692, 9.358195073339816359510978406405, 10.87154222802799952095100281738, 11.48745829338252566598719244875, 12.34987084576289245003522121718, 14.88777182222960426728115847717, 15.65123686670268935886069579282
