L(s) = 1 | + (−0.0977 + 0.169i)5-s + (2.48 + 0.900i)7-s + (1.54 − 0.890i)11-s + (5.11 + 2.95i)13-s + 0.588·17-s + 2.48i·19-s + (3.85 + 2.22i)23-s + (2.48 + 4.29i)25-s + (−6.28 + 3.62i)29-s + (−3.61 − 2.08i)31-s + (−0.395 + 0.333i)35-s − 2.57·37-s + (0.311 − 0.540i)41-s + (−5.08 − 8.80i)43-s + (−3.57 − 6.19i)47-s + ⋯ |
L(s) = 1 | + (−0.0437 + 0.0757i)5-s + (0.940 + 0.340i)7-s + (0.464 − 0.268i)11-s + (1.41 + 0.819i)13-s + 0.142·17-s + 0.570i·19-s + (0.804 + 0.464i)23-s + (0.496 + 0.859i)25-s + (−1.16 + 0.673i)29-s + (−0.649 − 0.374i)31-s + (−0.0668 + 0.0563i)35-s − 0.422·37-s + (0.0487 − 0.0843i)41-s + (−0.775 − 1.34i)43-s + (−0.521 − 0.903i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.269440785\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.269440785\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.48 - 0.900i)T \) |
good | 5 | \( 1 + (0.0977 - 0.169i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.54 + 0.890i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.11 - 2.95i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 0.588T + 17T^{2} \) |
| 19 | \( 1 - 2.48iT - 19T^{2} \) |
| 23 | \( 1 + (-3.85 - 2.22i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (6.28 - 3.62i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.61 + 2.08i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.57T + 37T^{2} \) |
| 41 | \( 1 + (-0.311 + 0.540i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.08 + 8.80i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.57 + 6.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 + (-5.75 + 9.97i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.57 + 3.79i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.927 - 1.60i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.58iT - 71T^{2} \) |
| 73 | \( 1 - 5.66iT - 73T^{2} \) |
| 79 | \( 1 + (2.92 + 5.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.740 + 1.28i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 9.44T + 89T^{2} \) |
| 97 | \( 1 + (-0.0722 + 0.0417i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.915631371021099158475670812483, −8.181528355931212288937269983036, −7.29374924061062155506651777863, −6.62538507603039195624231345373, −5.63163854774617643533654744839, −5.14465623037521316450408528294, −3.91887233456268243910041268713, −3.45819916751489454971453536005, −1.95735462394464739689971444713, −1.27314686689866948356544138531,
0.815217302472283630426351411156, 1.75892183845232795413612528405, 3.03145701937758145809393562690, 3.93737491007684407507064712202, 4.72799755592790763062628198968, 5.50444029340302407941889284225, 6.39455685351454339608036169609, 7.13719998342165190234921603862, 8.017497832601692898248639107397, 8.513706013184255823430596237379