L(s) = 1 | + (−0.866 + 0.5i)3-s + (1.73 − 2i)7-s + (−1 + 1.73i)9-s + (1.5 + 2.59i)11-s − 2i·13-s + (2.59 − 1.5i)17-s + (−0.5 + 0.866i)19-s + (−0.499 + 2.59i)21-s + (2.59 + 1.5i)23-s − 5i·27-s + 6·29-s + (3.5 + 6.06i)31-s + (−2.59 − 1.5i)33-s + (0.866 + 0.5i)37-s + (1 + 1.73i)39-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.288i)3-s + (0.654 − 0.755i)7-s + (−0.333 + 0.577i)9-s + (0.452 + 0.783i)11-s − 0.554i·13-s + (0.630 − 0.363i)17-s + (−0.114 + 0.198i)19-s + (−0.109 + 0.566i)21-s + (0.541 + 0.312i)23-s − 0.962i·27-s + 1.11·29-s + (0.628 + 1.08i)31-s + (−0.452 − 0.261i)33-s + (0.142 + 0.0821i)37-s + (0.160 + 0.277i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34322 + 0.312241i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34322 + 0.312241i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
good | 3 | \( 1 + (0.866 - 0.5i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (-2.59 + 1.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (-7.79 - 4.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.59 - 1.5i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.06 - 3.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-0.866 + 0.5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−10.46316978812784146183132950551, −9.966506999469627039730863512299, −8.744523783228056246188299062794, −7.81523461678017251021654824563, −7.11819984222519069098002794999, −5.92711842536935524451389851746, −4.94392405431303223645198880202, −4.29848960580906198104923715898, −2.83416821354201037765539083348, −1.20192676082104167361428121210,
0.995514160080586106266598406927, 2.54908013316194272179441710969, 3.86496266686828701105150565772, 5.10325235297456487298999083648, 6.00075083894100920395407015869, 6.61471578461042561160903581751, 7.86366889247894960880346816183, 8.730194098665843714287815495433, 9.330331143770651479197706482952, 10.58228229690028579587625957511