L(s) = 1 | + (0.411 + 0.411i)2-s − 1.66i·4-s + (0.180 + 0.672i)5-s + (2.60 − 0.483i)7-s + (1.50 − 1.50i)8-s + (−0.202 + 0.351i)10-s + (0.230 + 0.859i)11-s + (0.659 − 3.54i)13-s + (1.27 + 0.871i)14-s − 2.08·16-s + 0.0460·17-s + (−0.843 − 0.225i)19-s + (1.11 − 0.299i)20-s + (−0.258 + 0.448i)22-s + 3.19i·23-s + ⋯ |
L(s) = 1 | + (0.291 + 0.291i)2-s − 0.830i·4-s + (0.0806 + 0.300i)5-s + (0.983 − 0.182i)7-s + (0.532 − 0.532i)8-s + (−0.0641 + 0.111i)10-s + (0.0694 + 0.259i)11-s + (0.183 − 0.983i)13-s + (0.339 + 0.233i)14-s − 0.520·16-s + 0.0111·17-s + (−0.193 − 0.0518i)19-s + (0.249 − 0.0669i)20-s + (−0.0552 + 0.0956i)22-s + 0.665i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95985 - 0.619012i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95985 - 0.619012i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.60 + 0.483i)T \) |
| 13 | \( 1 + (-0.659 + 3.54i)T \) |
good | 2 | \( 1 + (-0.411 - 0.411i)T + 2iT^{2} \) |
| 5 | \( 1 + (-0.180 - 0.672i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.230 - 0.859i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 - 0.0460T + 17T^{2} \) |
| 19 | \( 1 + (0.843 + 0.225i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 - 3.19iT - 23T^{2} \) |
| 29 | \( 1 + (4.08 + 7.07i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.90 - 1.04i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (4.97 - 4.97i)T - 37iT^{2} \) |
| 41 | \( 1 + (-8.56 - 2.29i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.29 - 0.746i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-12.1 + 3.24i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (4.89 + 8.47i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.25 + 6.25i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.877 + 0.506i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.42 + 0.648i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.798 + 0.213i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (4.09 - 15.2i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (4.73 - 8.20i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.50 + 3.50i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.69 - 3.69i)T + 89iT^{2} \) |
| 97 | \( 1 + (0.288 + 1.07i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−10.26895040831491582857682797195, −9.450124591832689356097275757157, −8.307483285843874549219516935883, −7.52920590092175376487730442193, −6.58570367346839296579391859663, −5.67147110491883298765731761504, −4.92869896146895558621293974831, −3.96339290154788302006328207841, −2.39294332396832906180719843729, −1.05805616226933395218640104386,
1.60386442257045306505743895067, 2.75884132940561902223608779580, 4.05602368163807505702107118188, 4.69874998616149524514309352293, 5.76704930111700747076006617728, 7.05833046295088006236522543232, 7.75504791826990167513818591468, 8.887226010000870971699933729711, 8.997572330499636258102781620668, 10.82120191338954881641263683334