L(s) = 1 | + (−0.987 − 0.156i)2-s + (2.24 − 1.14i)3-s + (0.951 + 0.309i)4-s + (−0.898 − 2.04i)5-s + (−2.40 + 0.780i)6-s + (−1.82 + 1.91i)7-s + (−0.891 − 0.453i)8-s + (1.98 − 2.72i)9-s + (0.566 + 2.16i)10-s + (3.41 − 2.48i)11-s + (2.49 − 0.394i)12-s + (−0.770 − 4.86i)13-s + (2.09 − 1.60i)14-s + (−4.36 − 3.57i)15-s + (0.809 + 0.587i)16-s + (4.18 + 2.13i)17-s + ⋯ |
L(s) = 1 | + (−0.698 − 0.110i)2-s + (1.29 − 0.661i)3-s + (0.475 + 0.154i)4-s + (−0.401 − 0.915i)5-s + (−0.980 + 0.318i)6-s + (−0.688 + 0.725i)7-s + (−0.315 − 0.160i)8-s + (0.660 − 0.909i)9-s + (0.179 + 0.684i)10-s + (1.03 − 0.748i)11-s + (0.719 − 0.114i)12-s + (−0.213 − 1.34i)13-s + (0.561 − 0.430i)14-s + (−1.12 − 0.923i)15-s + (0.202 + 0.146i)16-s + (1.01 + 0.517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0355 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0355 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.943270 - 0.910280i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.943270 - 0.910280i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.987 + 0.156i)T \) |
| 5 | \( 1 + (0.898 + 2.04i)T \) |
| 7 | \( 1 + (1.82 - 1.91i)T \) |
good | 3 | \( 1 + (-2.24 + 1.14i)T + (1.76 - 2.42i)T^{2} \) |
| 11 | \( 1 + (-3.41 + 2.48i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.770 + 4.86i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-4.18 - 2.13i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (2.18 + 6.73i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.306 - 1.93i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.31 - 0.427i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.19 - 0.711i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.59 - 10.0i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-3.30 + 4.54i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (2.96 - 2.96i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.385 + 0.756i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-1.56 - 3.07i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-10.8 - 7.87i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.00 - 4.13i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.23 + 8.32i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (2.32 - 7.14i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.83 + 0.290i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-14.9 - 4.84i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (6.25 - 12.2i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.120 + 0.0874i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.57 + 3.09i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.38094721506735877832083320420, −9.945051330560914386081017264604, −9.062922395001559790302714320668, −8.547470938145928417387637863561, −7.895672236731211698648813681998, −6.77942775497913982973727759123, −5.52091472471529143593578188226, −3.60630952852457933808845636886, −2.70540713425226785473395583889, −1.06459642846216289516016069352,
2.13765860880447431896519547945, 3.55770754173970635596498605763, 4.12067977638584112893019803246, 6.37642722172071636113352725335, 7.19361942885338144632100142249, 7.977736433017119867134119715304, 9.148969855627173670548264859283, 9.786943654836472510178295291806, 10.31131650121096091161590015564, 11.52173208589997734742708150364