L(s) = 1 | + (−1.30 − 1.13i)3-s + (2.17 − 0.312i)7-s + (0.426 + 2.96i)9-s + (2.69 + 1.23i)13-s + (−0.372 + 2.59i)19-s + (−3.20 − 2.05i)21-s + (−2.07 + 4.54i)25-s + (2.80 − 4.37i)27-s + (9.91 − 4.53i)31-s + 10.4·37-s + (−2.13 − 4.67i)39-s + (3.26 − 11.1i)43-s + (−2.08 + 0.612i)49-s + (3.43 − 2.97i)57-s + (3.26 − 5.08i)61-s + ⋯ |
L(s) = 1 | + (−0.755 − 0.654i)3-s + (0.821 − 0.118i)7-s + (0.142 + 0.989i)9-s + (0.748 + 0.341i)13-s + (−0.0855 + 0.595i)19-s + (−0.698 − 0.448i)21-s + (−0.415 + 0.909i)25-s + (0.540 − 0.841i)27-s + (1.78 − 0.813i)31-s + 1.71·37-s + (−0.341 − 0.748i)39-s + (0.498 − 1.69i)43-s + (−0.297 + 0.0874i)49-s + (0.454 − 0.393i)57-s + (0.418 − 0.650i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.32601 - 0.268982i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.32601 - 0.268982i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.30 + 1.13i)T \) |
| 67 | \( 1 + (-1.74 + 7.99i)T \) |
good | 5 | \( 1 + (2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (-2.17 + 0.312i)T + (6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.69 - 1.23i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (0.372 - 2.59i)T + (-18.2 - 5.35i)T^{2} \) |
| 23 | \( 1 + (3.27 + 22.7i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (-9.91 + 4.53i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-3.26 + 11.1i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (6.68 + 46.5i)T^{2} \) |
| 53 | \( 1 + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-3.26 + 5.08i)T + (-25.3 - 55.4i)T^{2} \) |
| 71 | \( 1 + (-59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-0.774 - 0.498i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-9.85 - 4.50i)T + (51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 - 19.6iT - 97T^{2} \) |
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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
−10.43195793453150343684118244026, −9.395496855167589148268048616601, −8.185979718128660293964564320581, −7.74236432775987678710976311101, −6.62588832948977056444106016090, −5.89249941206047771508833267733, −4.94406861077144813704548522377, −3.94264313486251013822830128992, −2.23224949719835760957542690367, −1.07511815008898047306910656026,
1.03873672709058412323974492871, 2.82861639620989089966713772738, 4.21217971331135080300685429032, 4.83802919600695201299074283642, 5.90195949375533966161263024699, 6.58176593488884641425402084735, 7.895971926026955577371411017239, 8.617412384525111709091177701780, 9.634943250184338704037280649067, 10.38087731563022752781114895624