L(s) = 1 | + (−4.05 − 15.4i)2-s + 75.2·3-s + (−223. + 125. i)4-s + (−200. + 591. i)5-s + (−305. − 1.16e3i)6-s − 4.41e3·7-s + (2.85e3 + 2.94e3i)8-s − 900.·9-s + (9.97e3 + 706. i)10-s − 330. i·11-s + (−1.67e4 + 9.45e3i)12-s + 2.67e4i·13-s + (1.79e4 + 6.82e4i)14-s + (−1.51e4 + 4.45e4i)15-s + (3.39e4 − 5.60e4i)16-s + 6.86e4i·17-s + ⋯ |
L(s) = 1 | + (−0.253 − 0.967i)2-s + 0.928·3-s + (−0.871 + 0.490i)4-s + (−0.321 + 0.946i)5-s + (−0.235 − 0.898i)6-s − 1.83·7-s + (0.695 + 0.718i)8-s − 0.137·9-s + (0.997 + 0.0706i)10-s − 0.0225i·11-s + (−0.809 + 0.455i)12-s + 0.935i·13-s + (0.466 + 1.77i)14-s + (−0.298 + 0.879i)15-s + (0.518 − 0.855i)16-s + 0.822i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.346886 + 0.418166i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.346886 + 0.418166i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (4.05 + 15.4i)T \) |
| 5 | \( 1 + (200. - 591. i)T \) |
good | 3 | \( 1 - 75.2T + 6.56e3T^{2} \) |
| 7 | \( 1 + 4.41e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 330. iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.67e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 6.86e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.94e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 2.48e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 4.96e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 7.34e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.92e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 8.79e5T + 7.98e12T^{2} \) |
| 43 | \( 1 + 1.87e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 5.21e5T + 2.38e13T^{2} \) |
| 53 | \( 1 + 8.82e5iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 3.24e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 9.76e5T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.58e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.19e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 4.92e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 5.67e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 3.12e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 4.74e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 2.28e7iT - 7.83e15T^{2} \) |
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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
−16.96348729529571984194256305167, −15.33761219173127129751610100751, −13.92559678482826492492454489531, −12.92176027598462821312171265749, −11.29729721351060247136674429222, −9.850763663587515888564863136569, −8.795118778316551151316277823943, −6.87595400631418981730090806975, −3.59561074113976697561166499226, −2.66456566062306442112993230518,
0.26049630563938995218920607552, 3.53075838706728455187174832398, 5.74625963894324724843658260020, 7.59814362846174515755917191941, 8.909399630128673725610622771734, 9.808969810215638791379965635059, 12.68392152991986388093296747074, 13.54000828728433560385165860154, 15.04922682795661274782868038630, 16.10262694293482943351536375253