L(s) = 1 | + (40.5 − 23.3i)3-s + (−203. − 117. i)5-s + (1.13e3 − 2.11e3i)7-s + (1.09e3 − 1.89e3i)9-s + (−7.47e3 − 1.29e4i)11-s + 3.96e4i·13-s − 1.09e4·15-s + (8.46e4 − 4.88e4i)17-s + (−1.77e5 − 1.02e5i)19-s + (−3.75e3 − 1.12e5i)21-s + (−1.85e5 + 3.20e5i)23-s + (−1.67e5 − 2.90e5i)25-s − 1.02e5i·27-s + 1.17e5·29-s + (4.37e5 − 2.52e5i)31-s + ⋯ |
L(s) = 1 | + (0.5 − 0.288i)3-s + (−0.325 − 0.187i)5-s + (0.470 − 0.882i)7-s + (0.166 − 0.288i)9-s + (−0.510 − 0.884i)11-s + 1.38i·13-s − 0.216·15-s + (1.01 − 0.585i)17-s + (−1.36 − 0.787i)19-s + (−0.0192 − 0.577i)21-s + (−0.662 + 1.14i)23-s + (−0.429 − 0.743i)25-s − 0.192i·27-s + 0.165·29-s + (0.474 − 0.273i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.263348 - 1.20515i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.263348 - 1.20515i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-40.5 + 23.3i)T \) |
| 7 | \( 1 + (-1.13e3 + 2.11e3i)T \) |
good | 5 | \( 1 + (203. + 117. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (7.47e3 + 1.29e4i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 3.96e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-8.46e4 + 4.88e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.77e5 + 1.02e5i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.85e5 - 3.20e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.17e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-4.37e5 + 2.52e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (9.86e5 - 1.70e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 3.25e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 4.13e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (4.79e6 + 2.76e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (3.39e6 + 5.87e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.91e7 - 1.10e7i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (3.97e6 + 2.29e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.86e7 + 3.22e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 4.19e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.08e7 + 6.27e6i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (2.01e7 - 3.48e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 5.31e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-6.45e7 - 3.72e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 3.76e7iT - 7.83e15T^{2} \) |
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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
−12.10240758772438711162577874552, −11.17828061232939538937747286025, −9.915735260526479936025146620749, −8.579174844520238541663675938975, −7.71065530431830826583573388091, −6.51344523625507752583855959425, −4.73436574020099090617118892686, −3.51195848019717665719003001215, −1.81458989058477899385548176106, −0.32687356673926557475539624762,
1.88147194801375561251965987291, 3.18745112793570988220042828786, 4.69671046503967004155489800425, 5.98801221173737685730738082230, 7.82177619890387613070670721033, 8.367664576057048565960024095996, 9.912832896795858650613878326149, 10.71260580893343809369164198542, 12.24306013883972703499545907286, 12.86730082525955245742318590141