| L(s) = 1 | + (−1.99 + 0.407i)2-s + (1.34 + 0.217i)3-s + (1.97 − 0.842i)4-s + (−0.647 − 2.62i)5-s + (−2.76 + 0.111i)6-s + (−3.12 − 1.48i)7-s + (−0.250 + 0.172i)8-s + (−1.09 − 0.365i)9-s + (2.36 + 4.97i)10-s + (−0.457 − 1.36i)11-s + (2.83 − 0.698i)12-s + (−2.59 + 2.50i)13-s + (6.84 + 1.68i)14-s + (−0.295 − 3.65i)15-s + (−2.54 + 2.65i)16-s + (3.18 − 6.71i)17-s + ⋯ |
| L(s) = 1 | + (−1.41 + 0.288i)2-s + (0.773 + 0.125i)3-s + (0.988 − 0.421i)4-s + (−0.289 − 1.17i)5-s + (−1.12 + 0.0454i)6-s + (−1.18 − 0.560i)7-s + (−0.0885 + 0.0611i)8-s + (−0.365 − 0.121i)9-s + (0.746 + 1.57i)10-s + (−0.137 − 0.412i)11-s + (0.818 − 0.201i)12-s + (−0.718 + 0.695i)13-s + (1.82 + 0.450i)14-s + (−0.0762 − 0.944i)15-s + (−0.637 + 0.663i)16-s + (0.772 − 1.62i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0700 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0700 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.322060 - 0.345471i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.322060 - 0.345471i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.59 - 2.50i)T \) |
| good | 2 | \( 1 + (1.99 - 0.407i)T + (1.83 - 0.783i)T^{2} \) |
| 3 | \( 1 + (-1.34 - 0.217i)T + (2.84 + 0.950i)T^{2} \) |
| 5 | \( 1 + (0.647 + 2.62i)T + (-4.42 + 2.32i)T^{2} \) |
| 7 | \( 1 + (3.12 + 1.48i)T + (4.42 + 5.42i)T^{2} \) |
| 11 | \( 1 + (0.457 + 1.36i)T + (-8.79 + 6.60i)T^{2} \) |
| 17 | \( 1 + (-3.18 + 6.71i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (-1.35 + 0.783i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.33 + 4.04i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.83 - 8.97i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (-2.10 + 4.00i)T + (-17.6 - 25.5i)T^{2} \) |
| 37 | \( 1 + (-2.23 - 3.52i)T + (-15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (0.705 - 4.34i)T + (-38.8 - 12.9i)T^{2} \) |
| 43 | \( 1 + (-6.97 - 4.41i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (4.62 - 0.562i)T + (45.6 - 11.2i)T^{2} \) |
| 53 | \( 1 + (6.04 + 8.76i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (-2.94 + 2.82i)T + (2.37 - 58.9i)T^{2} \) |
| 61 | \( 1 + (1.14 + 0.0923i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (-2.26 + 5.30i)T + (-46.4 - 48.3i)T^{2} \) |
| 71 | \( 1 + (5.41 + 4.42i)T + (14.2 + 69.5i)T^{2} \) |
| 73 | \( 1 + (-0.379 + 0.428i)T + (-8.79 - 72.4i)T^{2} \) |
| 79 | \( 1 + (-0.816 - 6.72i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (1.28 + 0.488i)T + (62.1 + 55.0i)T^{2} \) |
| 89 | \( 1 + (-13.8 - 8.00i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.48 - 0.430i)T + (81.9 - 51.8i)T^{2} \) |
| 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
−12.54564637276778288305908266707, −11.33505295143201825239331527698, −9.826463161415089185926920665879, −9.397395929177219263137164610295, −8.626685267992241114051849341928, −7.67973331115333934645102539843, −6.65091830931659378667856634685, −4.75690352084653212600529660490, −3.06325715258386003847230237213, −0.61976259606704120967196323977,
2.41985480893920638490651372576, 3.31930093741477234729881203435, 5.92382972322012607896022497665, 7.32603132817230540219633548341, 7.975922344004189746330169300017, 9.085783994313366618720973752885, 9.985262905971005014247047506514, 10.59022378885663659784914102172, 11.81800091953986015157766354021, 12.90698178843920932963708227732
