| L(s) = 1 | + (5.65 − 9.79i)2-s + (−63.9 − 110. i)4-s + (−814. − 470. i)5-s + (14.7 − 2.40e3i)7-s − 1.44e3·8-s + (−9.21e3 + 5.32e3i)10-s + (−522. − 904. i)11-s − 4.70e4i·13-s + (−2.34e4 − 1.37e4i)14-s + (−8.19e3 + 1.41e4i)16-s + (−5.54e4 + 3.20e4i)17-s + (−2.03e5 − 1.17e5i)19-s + 1.20e5i·20-s − 1.18e4·22-s + (1.35e5 − 2.35e5i)23-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−1.30 − 0.752i)5-s + (0.00616 − 0.999i)7-s − 0.353·8-s + (−0.921 + 0.532i)10-s + (−0.0356 − 0.0617i)11-s − 1.64i·13-s + (−0.610 − 0.357i)14-s + (−0.125 + 0.216i)16-s + (−0.664 + 0.383i)17-s + (−1.56 − 0.901i)19-s + 0.752i·20-s − 0.0504·22-s + (0.485 − 0.840i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 - 0.991i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.6523973702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6523973702\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.65 + 9.79i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-14.7 + 2.40e3i)T \) |
| good | 5 | \( 1 + (814. + 470. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (522. + 904. i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 4.70e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (5.54e4 - 3.20e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (2.03e5 + 1.17e5i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-1.35e5 + 2.35e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.26e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-1.43e5 + 8.28e4i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (1.35e6 - 2.34e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 1.48e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 9.28e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-8.09e6 - 4.67e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-4.62e5 - 8.01e5i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.14e7 - 6.61e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (1.21e7 + 7.02e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-8.70e6 - 1.50e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 5.33e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-2.95e7 + 1.70e7i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.61e7 + 2.79e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 6.53e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-2.58e7 - 1.49e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + 5.40e7iT - 7.83e15T^{2} \) |
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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
−10.88971269247686105687008652041, −10.47027414165183885378765506363, −8.718753398378188409836082015874, −7.997539224193925686478955985699, −6.60971674519880712568021741037, −4.81759012499282380121546705292, −4.17143383638676010480213047801, −2.89548643421147788839303629546, −0.902251750286684671437840430923, −0.19665775904694909563214755216,
2.27493150690753956546115254748, 3.69673962117958621180957220430, 4.68060278297808447836190748912, 6.29723622823338822713306223258, 7.07241499126847393805270823696, 8.273314873272087517363815418141, 9.119410480998798569685391491213, 10.80569306795023065375707569125, 11.80300368869170010528197749742, 12.36663486745292953833355555197
