L(s) = 1 | + (1.48 − 1.67i)5-s + (0.732 + 2.73i)7-s + (−2.44 − 1.41i)11-s + (1.09 − 4.09i)13-s + (−1.41 − 1.41i)17-s − 4i·19-s + (−7.72 − 2.07i)23-s + (−0.598 − 4.96i)25-s + (4.94 − 8.57i)29-s + (4 + 6.92i)31-s + (5.65 + 2.82i)35-s + (−3 + 3i)37-s + (−1.22 + 0.707i)41-s + (11.5 − 3.10i)47-s + (−0.866 + 0.5i)49-s + ⋯ |
L(s) = 1 | + (0.663 − 0.748i)5-s + (0.276 + 1.03i)7-s + (−0.738 − 0.426i)11-s + (0.304 − 1.13i)13-s + (−0.342 − 0.342i)17-s − 0.917i·19-s + (−1.61 − 0.431i)23-s + (−0.119 − 0.992i)25-s + (0.919 − 1.59i)29-s + (0.718 + 1.24i)31-s + (0.956 + 0.478i)35-s + (−0.493 + 0.493i)37-s + (−0.191 + 0.110i)41-s + (1.69 − 0.453i)47-s + (−0.123 + 0.0714i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.620344357\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.620344357\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.48 + 1.67i)T \) |
good | 7 | \( 1 + (-0.732 - 2.73i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (2.44 + 1.41i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.09 + 4.09i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (1.41 + 1.41i)T + 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (7.72 + 2.07i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-4.94 + 8.57i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3 - 3i)T - 37iT^{2} \) |
| 41 | \( 1 + (1.22 - 0.707i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-11.5 + 3.10i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-7.07 + 7.07i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.41 + 2.44i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.9 + 2.92i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-7 - 7i)T + 73iT^{2} \) |
| 79 | \( 1 + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.10 + 11.5i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 - 1.41T + 89T^{2} \) |
| 97 | \( 1 + (-1.09 - 4.09i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−9.030631217075662760350577090489, −8.437884864803261458017446068933, −7.978676211688991537880721225193, −6.59583320780520481091520519747, −5.75851804842946926497475128344, −5.26638090719120149575292530192, −4.37398918794355801217143555273, −2.86270285547027462086645026550, −2.18275851933252993978176173930, −0.61630672688956756628791821058,
1.51453008937336108994909454266, 2.43331521916558399397102460263, 3.76458708539468814704926496897, 4.42113655001871931624261668237, 5.65192800827658845894220903689, 6.37312388379521081659495655142, 7.23042361099159651778753093407, 7.78470146965479903874083742422, 8.843037765495094867790385547061, 9.744137819803175964166276339339