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
−10.38087731563022752781114895624, −9.634943250184338704037280649067, −8.617412384525111709091177701780, −7.895971926026955577371411017239, −6.58176593488884641425402084735, −5.90195949375533966161263024699, −4.83802919600695201299074283642, −4.21217971331135080300685429032, −2.82861639620989089966713772738, −1.03873672709058412323974492871,
1.07511815008898047306910656026, 2.23224949719835760957542690367, 3.94264313486251013822830128992, 4.94406861077144813704548522377, 5.89249941206047771508833267733, 6.62588832948977056444106016090, 7.74236432775987678710976311101, 8.185979718128660293964564320581, 9.395496855167589148268048616601, 10.43195793453150343684118244026