L(s) = 1 | + (−10.1 + 5.01i)2-s + (8.25 − 8.25i)3-s + (77.7 − 101. i)4-s + (63.0 + 63.0i)5-s + (−42.3 + 125. i)6-s + 847. i·7-s + (−277. + 1.42e3i)8-s + 2.05e3i·9-s + (−955. − 323. i)10-s + (2.94e3 + 2.94e3i)11-s + (−198. − 1.48e3i)12-s + (910. − 910. i)13-s + (−4.25e3 − 8.59e3i)14-s + 1.04e3·15-s + (−4.30e3 − 1.58e4i)16-s + 1.68e4·17-s + ⋯ |
L(s) = 1 | + (−0.896 + 0.443i)2-s + (0.176 − 0.176i)3-s + (0.607 − 0.794i)4-s + (0.225 + 0.225i)5-s + (−0.0799 + 0.236i)6-s + 0.934i·7-s + (−0.191 + 0.981i)8-s + 0.937i·9-s + (−0.302 − 0.102i)10-s + (0.666 + 0.666i)11-s + (−0.0331 − 0.247i)12-s + (0.114 − 0.114i)13-s + (−0.414 − 0.837i)14-s + 0.0796·15-s + (−0.262 − 0.964i)16-s + 0.834·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.804624 + 0.708717i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.804624 + 0.708717i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (10.1 - 5.01i)T \) |
good | 3 | \( 1 + (-8.25 + 8.25i)T - 2.18e3iT^{2} \) |
| 5 | \( 1 + (-63.0 - 63.0i)T + 7.81e4iT^{2} \) |
| 7 | \( 1 - 847. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + (-2.94e3 - 2.94e3i)T + 1.94e7iT^{2} \) |
| 13 | \( 1 + (-910. + 910. i)T - 6.27e7iT^{2} \) |
| 17 | \( 1 - 1.68e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + (1.10e4 - 1.10e4i)T - 8.93e8iT^{2} \) |
| 23 | \( 1 - 6.76e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + (-7.27e4 + 7.27e4i)T - 1.72e10iT^{2} \) |
| 31 | \( 1 + 2.43e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (1.11e5 + 1.11e5i)T + 9.49e10iT^{2} \) |
| 41 | \( 1 + 8.59e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (-1.08e5 - 1.08e5i)T + 2.71e11iT^{2} \) |
| 47 | \( 1 + 3.87e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-1.29e6 - 1.29e6i)T + 1.17e12iT^{2} \) |
| 59 | \( 1 + (-1.70e6 - 1.70e6i)T + 2.48e12iT^{2} \) |
| 61 | \( 1 + (1.95e6 - 1.95e6i)T - 3.14e12iT^{2} \) |
| 67 | \( 1 + (-3.23e6 + 3.23e6i)T - 6.06e12iT^{2} \) |
| 71 | \( 1 + 3.07e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 2.72e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 2.89e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + (-2.74e6 + 2.74e6i)T - 2.71e13iT^{2} \) |
| 89 | \( 1 + 2.91e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 8.03e6T + 8.07e13T^{2} \) |
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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
−17.87649158366379185896230130709, −16.60067381697358894070157132755, −15.29767332958949621642025525274, −14.10466550100659282853131204924, −12.06412922231822754305225082076, −10.39855908493668988509854056821, −8.964133635084294076231536063377, −7.48135263518803584754621551438, −5.69242039416159554140727523505, −2.01483339141988993441643442171,
0.943474408132383081074578198487, 3.59963826862410710046741107350, 6.76697461607006599077354336057, 8.615723467382185798111833795494, 9.879425485033139466996918909673, 11.30366661571238781698052932524, 12.82352006380262637450429517218, 14.54673679656296112732637988177, 16.35683337638688442015840697901, 17.19902143275778776449386769923