L(s) = 1 | + (−10 − 12.4i)2-s − 99.9i·3-s + (−56 + 249. i)4-s + 610·5-s + (−1.24e3 + 999. i)6-s − 1.39e3i·7-s + (3.68e3 − 1.79e3i)8-s − 3.42e3·9-s + (−6.10e3 − 7.61e3i)10-s + 1.84e4i·11-s + (2.49e4 + 5.59e3i)12-s − 5.47e3·13-s + (−1.74e4 + 1.39e4i)14-s − 6.09e4i·15-s + (−5.92e4 − 2.79e4i)16-s + 7.30e4·17-s + ⋯ |
L(s) = 1 | + (−0.625 − 0.780i)2-s − 1.23i·3-s + (−0.218 + 0.975i)4-s + 0.976·5-s + (−0.962 + 0.770i)6-s − 0.582i·7-s + (0.898 − 0.439i)8-s − 0.521·9-s + (−0.609 − 0.761i)10-s + 1.26i·11-s + (1.20 + 0.269i)12-s − 0.191·13-s + (−0.454 + 0.364i)14-s − 1.20i·15-s + (−0.904 − 0.426i)16-s + 0.875·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.608015 - 0.759411i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.608015 - 0.759411i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (10 + 12.4i)T \) |
good | 3 | \( 1 + 99.9iT - 6.56e3T^{2} \) |
| 5 | \( 1 - 610T + 3.90e5T^{2} \) |
| 7 | \( 1 + 1.39e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.84e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 5.47e3T + 8.15e8T^{2} \) |
| 17 | \( 1 - 7.30e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.94e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 2.37e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.28e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 6.79e4iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 3.47e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.14e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 5.92e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 7.62e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 8.24e5T + 6.22e13T^{2} \) |
| 59 | \( 1 - 3.72e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.47e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.52e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 1.19e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 5.72e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.59e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 5.19e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 8.33e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 1.20e8T + 7.83e15T^{2} \) |
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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
−23.05283787075017977522675009153, −21.15962427867806757942742231853, −19.64969778921455067094912119585, −18.15585329877628661872716835672, −17.23027320724048646915557601166, −13.63807897941524328856721670403, −12.33245402130425562647369855681, −9.931355639451609166898450403277, −7.36865938776434711337216230376, −1.67164860332379608344561487750,
5.61132311738355574390567068510, 9.012485997895990053334791366615, 10.42394208697097118220815736412, 14.18255958717057463814967712833, 15.77275031872313453029833508855, 17.02219167945562673525700051124, 18.79159807457104404394184152711, 21.04356257984729831813228989909, 22.24342160857562206932762835749, 24.38655660422666686374795816593