| L(s) = 1 | + (0.577 − 1.29i)2-s + (−1.73 − 0.0662i)3-s + (−1.33 − 1.49i)4-s + (−2.45 + 0.322i)5-s + (−1.08 + 2.19i)6-s + (1.23 − 0.331i)7-s + (−2.69 + 0.860i)8-s + (2.99 + 0.229i)9-s + (−0.999 + 3.35i)10-s + (−2.95 + 2.26i)11-s + (2.20 + 2.66i)12-s + (−1.91 + 2.50i)13-s + (0.286 − 1.79i)14-s + (4.26 − 0.396i)15-s + (−0.445 + 3.97i)16-s + 5.86i·17-s + ⋯ |
| L(s) = 1 | + (0.408 − 0.912i)2-s + (−0.999 − 0.0382i)3-s + (−0.666 − 0.745i)4-s + (−1.09 + 0.144i)5-s + (−0.442 + 0.896i)6-s + (0.468 − 0.125i)7-s + (−0.952 + 0.304i)8-s + (0.997 + 0.0764i)9-s + (−0.316 + 1.06i)10-s + (−0.890 + 0.683i)11-s + (0.637 + 0.770i)12-s + (−0.532 + 0.693i)13-s + (0.0766 − 0.478i)14-s + (1.10 − 0.102i)15-s + (−0.111 + 0.993i)16-s + 1.42i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0517932 + 0.0632495i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0517932 + 0.0632495i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.577 + 1.29i)T \) |
| 3 | \( 1 + (1.73 + 0.0662i)T \) |
| good | 5 | \( 1 + (2.45 - 0.322i)T + (4.82 - 1.29i)T^{2} \) |
| 7 | \( 1 + (-1.23 + 0.331i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (2.95 - 2.26i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (1.91 - 2.50i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 - 5.86iT - 17T^{2} \) |
| 19 | \( 1 + (1.91 + 4.62i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (4.05 + 1.08i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.249 + 1.89i)T + (-28.0 - 7.50i)T^{2} \) |
| 31 | \( 1 + (2.62 + 4.54i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.48 + 6.00i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (2.39 + 0.642i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-2.17 + 1.66i)T + (11.1 - 41.5i)T^{2} \) |
| 47 | \( 1 + (7.89 + 4.55i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.93 - 2.45i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (12.7 - 1.67i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (1.69 - 12.8i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-8.64 - 6.63i)T + (17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (1.70 + 1.70i)T + 71iT^{2} \) |
| 73 | \( 1 + (1.37 - 1.37i)T - 73iT^{2} \) |
| 79 | \( 1 + (-0.644 - 0.372i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-14.3 - 1.88i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (1.40 + 1.40i)T + 89iT^{2} \) |
| 97 | \( 1 + (2.32 - 4.02i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.01289036769021986840476893771, −11.18606918356977507201249287548, −10.63138485363081696846442887561, −9.657909366371428639492484194657, −8.200954700185246277615397391890, −7.15887297219059935739157647325, −5.84437770752594691778131695998, −4.60731498190174908701882708284, −4.05798366735315804826734128466, −2.07050883005437843358257496881,
0.05882389824584051744474871312, 3.40166836206070731786419515564, 4.73884589713314126467264543550, 5.34819555948084410581490561135, 6.52644805410874238844934785016, 7.79760152827103708959447767939, 8.042283567859372774147262441930, 9.636358764509545694554851580232, 10.83506637321388106172536146664, 11.81195306092219557346841507598
