| 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
−11.52173208589997734742708150364, −10.31131650121096091161590015564, −9.786943654836472510178295291806, −9.148969855627173670548264859283, −7.977736433017119867134119715304, −7.19361942885338144632100142249, −6.37642722172071636113352725335, −4.12067977638584112893019803246, −3.55770754173970635596498605763, −2.13765860880447431896519547945,
1.06459642846216289516016069352, 2.70540713425226785473395583889, 3.60630952852457933808845636886, 5.52091472471529143593578188226, 6.77942775497913982973727759123, 7.895672236731211698648813681998, 8.547470938145928417387637863561, 9.062922395001559790302714320668, 9.945051330560914386081017264604, 11.38094721506735877832083320420
