| L(s) = 1 | + (−0.909 − 0.415i)2-s + (0.537 − 1.83i)3-s + (0.654 + 0.755i)4-s + (−0.575 − 2.16i)5-s + (−1.24 + 1.44i)6-s + (2.93 + 0.422i)7-s + (−0.281 − 0.959i)8-s + (−0.537 − 0.345i)9-s + (−0.373 + 2.20i)10-s + (−1.22 − 2.68i)11-s + (1.73 − 0.792i)12-s + (−1.87 + 0.269i)13-s + (−2.49 − 1.60i)14-s + (−4.26 − 0.107i)15-s + (−0.142 + 0.989i)16-s + (−0.719 − 0.623i)17-s + ⋯ |
| L(s) = 1 | + (−0.643 − 0.293i)2-s + (0.310 − 1.05i)3-s + (0.327 + 0.377i)4-s + (−0.257 − 0.966i)5-s + (−0.510 + 0.588i)6-s + (1.10 + 0.159i)7-s + (−0.0996 − 0.339i)8-s + (−0.179 − 0.115i)9-s + (−0.118 + 0.697i)10-s + (−0.370 − 0.810i)11-s + (0.500 − 0.228i)12-s + (−0.520 + 0.0748i)13-s + (−0.666 − 0.428i)14-s + (−1.10 − 0.0276i)15-s + (−0.0355 + 0.247i)16-s + (−0.174 − 0.151i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.551249 - 0.860907i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.551249 - 0.860907i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.909 + 0.415i)T \) |
| 5 | \( 1 + (0.575 + 2.16i)T \) |
| 23 | \( 1 + (4.04 - 2.57i)T \) |
| good | 3 | \( 1 + (-0.537 + 1.83i)T + (-2.52 - 1.62i)T^{2} \) |
| 7 | \( 1 + (-2.93 - 0.422i)T + (6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (1.22 + 2.68i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (1.87 - 0.269i)T + (12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (0.719 + 0.623i)T + (2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-0.167 - 0.193i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (0.0268 - 0.0309i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-5.13 + 1.50i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-2.00 + 3.12i)T + (-15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-7.88 + 5.06i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (3.10 - 10.5i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 8.07iT - 47T^{2} \) |
| 53 | \( 1 + (-12.5 - 1.79i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.794 - 5.52i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-8.08 + 2.37i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-5.22 - 2.38i)T + (43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (5.95 - 13.0i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.785 + 0.680i)T + (10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.123 + 0.862i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-6.11 + 9.51i)T + (-34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (15.5 + 4.57i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-1.81 - 2.82i)T + (-40.2 + 88.2i)T^{2} \) |
| 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
−11.87449764057457869702596716119, −11.19393952859297797539353998669, −9.825915377687280563477026983987, −8.624884604224627044658457024977, −8.047877133586836142989669901566, −7.36750037272609800612540427122, −5.77354114319363324374866842496, −4.39809070898041421146449298385, −2.38755438825374750922148432984, −1.10571220615649600154034560936,
2.36188314423427414678488658420, 4.03785244677526596887703164172, 5.07916238812621050644549160759, 6.71154809621211891320509271772, 7.68466332756212994470221096951, 8.554111043443802134917141690350, 9.905638983709778995001254251412, 10.27872750653612637474791284284, 11.20041862818644878968776852204, 12.17754813723771146208374334826
