| L(s) = 1 | + (−0.156 − 0.987i)2-s + (1.37 − 2.69i)3-s + (−0.951 + 0.309i)4-s + (−2.09 + 0.773i)5-s + (−2.88 − 0.936i)6-s + (−2.43 + 1.04i)7-s + (0.453 + 0.891i)8-s + (−3.62 − 4.99i)9-s + (1.09 + 1.95i)10-s + (0.762 + 0.553i)11-s + (−0.473 + 2.99i)12-s + (−5.27 − 0.835i)13-s + (1.41 + 2.23i)14-s + (−0.795 + 6.72i)15-s + (0.809 − 0.587i)16-s + (−2.27 − 4.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 − 0.698i)2-s + (0.793 − 1.55i)3-s + (−0.475 + 0.154i)4-s + (−0.938 + 0.346i)5-s + (−1.17 − 0.382i)6-s + (−0.918 + 0.395i)7-s + (0.160 + 0.315i)8-s + (−1.20 − 1.66i)9-s + (0.345 + 0.616i)10-s + (0.229 + 0.167i)11-s + (−0.136 + 0.863i)12-s + (−1.46 − 0.231i)13-s + (0.377 + 0.597i)14-s + (−0.205 + 1.73i)15-s + (0.202 − 0.146i)16-s + (−0.551 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.171677 + 0.793006i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.171677 + 0.793006i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.156 + 0.987i)T \) |
| 5 | \( 1 + (2.09 - 0.773i)T \) |
| 7 | \( 1 + (2.43 - 1.04i)T \) |
| good | 3 | \( 1 + (-1.37 + 2.69i)T + (-1.76 - 2.42i)T^{2} \) |
| 11 | \( 1 + (-0.762 - 0.553i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (5.27 + 0.835i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (2.27 + 4.46i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.25 + 6.94i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.865 + 0.137i)T + (21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-2.91 + 0.948i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-7.79 - 2.53i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.78 - 0.916i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-1.71 - 2.36i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-1.59 + 1.59i)T - 43iT^{2} \) |
| 47 | \( 1 + (8.89 + 4.53i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (5.12 + 2.60i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (7.59 - 5.51i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.17 + 4.36i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (1.42 - 0.727i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-0.614 - 1.89i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.34 - 8.46i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (6.44 - 2.09i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (2.80 - 1.42i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (3.12 + 2.26i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-16.0 - 8.18i)T + (57.0 + 78.4i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.44041413122066674787769813516, −9.846734740841251743073680820866, −9.069344366675927787341830835835, −8.102465498958016840284663852435, −7.15699237126097386185817409839, −6.66769128730547560769147249827, −4.72748718854079290889282650902, −2.91175760683245927227574227851, −2.67165266193542322560204250171, −0.51715080331810623008932923414,
3.13501808487272806542623910532, 4.10202730756183278077444854393, 4.74690586949524837957793387243, 6.19930125571564211552826355351, 7.61859866437817439821317447529, 8.288643818555539383800321891931, 9.319933953770859432110224789396, 9.878522951069563291938609119973, 10.69042253789900909510077969671, 12.01197716024189456675321995079
