| L(s) = 1 | + (−0.826 − 0.563i)2-s + (2.12 + 1.97i)3-s + (0.365 + 0.930i)4-s + (3.39 − 1.04i)5-s + (−0.646 − 2.83i)6-s + (0.222 − 0.974i)8-s + (0.405 + 5.41i)9-s + (−3.39 − 1.04i)10-s + (−0.265 + 3.54i)11-s + (−1.06 + 2.70i)12-s + (−1.94 + 0.936i)13-s + (9.29 + 4.47i)15-s + (−0.733 + 0.680i)16-s + (−1.12 + 0.169i)17-s + (2.71 − 4.70i)18-s + (0.449 + 0.777i)19-s + ⋯ |
| L(s) = 1 | + (−0.584 − 0.398i)2-s + (1.22 + 1.14i)3-s + (0.182 + 0.465i)4-s + (1.51 − 0.468i)5-s + (−0.263 − 1.15i)6-s + (0.0786 − 0.344i)8-s + (0.135 + 1.80i)9-s + (−1.07 − 0.331i)10-s + (−0.0800 + 1.06i)11-s + (−0.306 + 0.780i)12-s + (−0.539 + 0.259i)13-s + (2.40 + 1.15i)15-s + (−0.183 + 0.170i)16-s + (−0.273 + 0.0411i)17-s + (0.640 − 1.10i)18-s + (0.103 + 0.178i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.95296 + 0.782963i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95296 + 0.782963i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.826 + 0.563i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-2.12 - 1.97i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-3.39 + 1.04i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.265 - 3.54i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (1.94 - 0.936i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (1.12 - 0.169i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.449 - 0.777i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.01 + 0.454i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (3.37 + 4.23i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-3.48 + 6.03i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.35 + 8.54i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-0.336 + 1.47i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-1.58 - 6.93i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (3.59 + 2.45i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (1.30 + 3.33i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-3.78 - 1.16i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (3.61 - 9.22i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-3.32 + 5.75i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.69 + 10.9i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (9.34 - 6.37i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-1.15 - 2.00i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.0 - 5.32i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.735 + 9.81i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + 2.24T + 97T^{2} \) |
| 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
−9.985878826947281025488621517618, −9.713373367579811199374237362003, −9.274950090493790531642506629176, −8.338156861511900778790215283761, −7.44234484316600990433059807760, −6.02955770782134921789182029375, −4.82092001030587347762436066444, −4.01294662146002991643562914297, −2.50354849464610004402100841272, −1.99090449894969029672030895778,
1.32653291862813189107002980770, 2.36788180898469717072257584346, 3.17022572375783744453245097757, 5.31461572713749186312073257796, 6.32068203118137498680813985098, 6.85316439904416940553137122788, 7.85786106307158929328061380909, 8.592001351635874703827106004424, 9.324068863542441304435556211892, 10.05061715501326536196097134783
