| L(s) = 1 | + 3.06i·2-s − 9.88·3-s + 118.·4-s + 42.5·5-s − 30.3i·6-s + 961. i·7-s + 756. i·8-s − 2.08e3·9-s + 130. i·10-s − 2.36e3i·11-s − 1.17e3·12-s + 1.10e4·13-s − 2.94e3·14-s − 420.·15-s + 1.28e4·16-s + 2.02e4i·17-s + ⋯ |
| L(s) = 1 | + 0.271i·2-s − 0.211·3-s + 0.926·4-s + 0.152·5-s − 0.0573i·6-s + 1.05i·7-s + 0.522i·8-s − 0.955·9-s + 0.0412i·10-s − 0.536i·11-s − 0.195·12-s + 1.39·13-s − 0.287·14-s − 0.0321·15-s + 0.784·16-s + 0.997i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.17934 + 1.47400i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17934 + 1.47400i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (3.88e5 + 1.72e6i)T \) |
| good | 2 | \( 1 - 3.06iT - 128T^{2} \) |
| 3 | \( 1 + 9.88T + 2.18e3T^{2} \) |
| 5 | \( 1 - 42.5T + 7.81e4T^{2} \) |
| 7 | \( 1 - 961. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 2.36e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.10e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.02e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 2.97e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.91e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 1.04e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 4.93e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 2.15e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 2.72e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.38e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 2.91e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.35e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.13e6iT - 2.48e12T^{2} \) |
| 67 | \( 1 + 3.25e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 5.52e5iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 2.12e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.09e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 1.00e7T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.71e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 4.79e6T + 8.07e13T^{2} \) |
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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
−14.04821500730885710341703898598, −12.63414810843063071884778906765, −11.44674750905931166228833989464, −10.85076116928987832812938746148, −8.970224361770814012983493290231, −8.021983353347097935778925477841, −6.12786671441551817197078100716, −5.78054306179687416147340262068, −3.28105562567648790796094182938, −1.76533313509115039226953489180,
0.68671684910844676450285248031, 2.40335116711847950279745191083, 4.02991466183027870000643236367, 5.98554000035194818924770128648, 7.01611118339111913611212002660, 8.436924330867487566980993398276, 10.15954918012522396367047818409, 10.98570998377921621054783247274, 11.88400984772077281742758807614, 13.23221550783502750748349027293
