| L(s) = 1 | + (−1.72 − 1.44i)2-s + (2.01 − 0.731i)3-s + (0.530 + 3.00i)4-s + (0.706 − 4.00i)5-s + (−4.51 − 1.64i)6-s + (0.415 + 0.720i)7-s + (1.18 − 2.05i)8-s + (1.20 − 1.01i)9-s + (−7.00 + 5.88i)10-s + (0.5 − 0.866i)11-s + (3.26 + 5.66i)12-s + (3.12 + 1.13i)13-s + (0.324 − 1.84i)14-s + (−1.51 − 8.57i)15-s + (0.725 − 0.263i)16-s + (−2.88 − 2.42i)17-s + ⋯ |
| L(s) = 1 | + (−1.21 − 1.02i)2-s + (1.16 − 0.422i)3-s + (0.265 + 1.50i)4-s + (0.316 − 1.79i)5-s + (−1.84 − 0.671i)6-s + (0.157 + 0.272i)7-s + (0.419 − 0.727i)8-s + (0.402 − 0.337i)9-s + (−2.21 + 1.86i)10-s + (0.150 − 0.261i)11-s + (0.943 + 1.63i)12-s + (0.866 + 0.315i)13-s + (0.0868 − 0.492i)14-s + (−0.390 − 2.21i)15-s + (0.181 − 0.0659i)16-s + (−0.699 − 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.369965 - 0.910162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.369965 - 0.910162i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (1.95 - 3.89i)T \) |
| good | 2 | \( 1 + (1.72 + 1.44i)T + (0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-2.01 + 0.731i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.706 + 4.00i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.415 - 0.720i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (-3.12 - 1.13i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.88 + 2.42i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.781 - 4.43i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.69 - 1.42i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (2.57 + 4.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.19T + 37T^{2} \) |
| 41 | \( 1 + (-10.6 + 3.88i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.07 - 6.07i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (1.73 - 1.45i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.825 - 4.67i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.32 - 5.30i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.459 + 2.60i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.33 + 6.15i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.47 - 8.35i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (8.22 - 2.99i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-11.9 + 4.33i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (3.23 + 5.61i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-12.5 - 4.55i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.556 - 0.466i)T + (16.8 + 95.5i)T^{2} \) |
| 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
−11.93052433663770714755644030784, −11.04588802620688332934291148033, −9.456201298353070139538632701335, −9.156716707471139944885868734781, −8.408974835299766653321300682368, −7.76628843401023245259898263103, −5.66647408492937695715357381444, −3.93606555980989507584732253420, −2.27471861084796283370758495374, −1.25935091838025787484033758280,
2.47506112634491080222784136633, 3.82001214292249061814701527471, 6.15111494072588852646543945892, 6.85632894756859935391214002075, 7.82855778290464859236107947827, 8.729444955976461034561801070965, 9.557630022475734877710432664492, 10.51775257214852625655876453905, 11.04471108395495880027148117557, 13.18172531752116579279727694964
