L(s) = 1 | + (1.5 − 0.866i)3-s + (−0.621 − 0.358i)5-s + (2.62 − 0.358i)7-s + (1.5 − 2.59i)9-s + (2.91 + 5.04i)11-s − 1.24·15-s + (3.62 − 2.80i)21-s + (−2.24 − 3.88i)25-s − 5.19i·27-s − 7.58·29-s + (9.62 − 5.55i)31-s + (8.74 + 5.04i)33-s + (−1.75 − 0.717i)35-s + (−1.86 + 1.07i)45-s + (6.74 − 1.88i)49-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s + (−0.277 − 0.160i)5-s + (0.990 − 0.135i)7-s + (0.5 − 0.866i)9-s + (0.878 + 1.52i)11-s − 0.320·15-s + (0.790 − 0.612i)21-s + (−0.448 − 0.776i)25-s − 0.999i·27-s − 1.40·29-s + (1.72 − 0.997i)31-s + (1.52 + 0.878i)33-s + (−0.297 − 0.121i)35-s + (−0.277 + 0.160i)45-s + (0.963 − 0.268i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.13124 - 0.560512i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.13124 - 0.560512i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.62 + 0.358i)T \) |
good | 5 | \( 1 + (0.621 + 0.358i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.91 - 5.04i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.58T + 29T^{2} \) |
| 31 | \( 1 + (-9.62 + 5.55i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.03 - 3.52i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (12.9 - 7.49i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (8.48 - 4.89i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.86 - 15.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.5iT - 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 8.06iT - 97T^{2} \) |
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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
−10.23259890897280968010745826422, −9.469588114412123537313569081449, −8.625764852447604011237988708515, −7.74265486223848597575391910580, −7.22980091920701204937054134655, −6.12696773276472184504705824930, −4.57401618642052231341101273731, −4.01888894305330788164185027004, −2.41005208670120327193575356706, −1.40667237725378205320687868986,
1.55416361376436812678291719203, 3.05061270706615394995318809565, 3.89317499024493505577282416227, 4.93670492334240298154299997817, 6.02922910124575637344521453271, 7.31379558293676093406956442827, 8.182330239286997046352751271829, 8.747151441455426521058949486021, 9.540203843719486853021143231231, 10.69007475041549822905849875446