L(s) = 1 | + 19.6i·2-s − 11.3·3-s − 259.·4-s + 547.·5-s − 223. i·6-s + 1.20e3i·7-s − 2.59e3i·8-s − 2.05e3·9-s + 1.07e4i·10-s + 1.97e3i·11-s + 2.94e3·12-s + 3.52e3·13-s − 2.36e4·14-s − 6.20e3·15-s + 1.78e4·16-s + 2.13e4i·17-s + ⋯ |
L(s) = 1 | + 1.74i·2-s − 0.242·3-s − 2.02·4-s + 1.95·5-s − 0.421i·6-s + 1.32i·7-s − 1.79i·8-s − 0.941·9-s + 3.40i·10-s + 0.446i·11-s + 0.491·12-s + 0.444·13-s − 2.30·14-s − 0.474·15-s + 1.08·16-s + 1.05i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.644057 - 1.46105i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.644057 - 1.46105i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 + (1.19e6 - 1.30e6i)T \) |
good | 2 | \( 1 - 19.6iT - 128T^{2} \) |
| 3 | \( 1 + 11.3T + 2.18e3T^{2} \) |
| 5 | \( 1 - 547.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.20e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 1.97e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 3.52e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.13e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 3.97e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.45e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 1.94e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 2.26e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 5.66e4iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 5.80e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.57e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 1.44e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.06e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 4.39e4iT - 2.48e12T^{2} \) |
| 67 | \( 1 - 1.61e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.27e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 4.52e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.82e5iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 8.04e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.67e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 4.82e6T + 8.07e13T^{2} \) |
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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
−14.62188368461741801658399888288, −13.52299186728949537891804304931, −12.56179716099156206702896331197, −10.47750680655939977298369631926, −9.023896723851727320853586629010, −8.589091115721507794448689484051, −6.43605762441590171271177792064, −5.96850292317948557733314319267, −5.09931170611910557612457092112, −2.18090555674253863584563097872,
0.58945456269561840188095448707, 1.84140019593219818111044991655, 3.16117071314091417179817942314, 4.91507761219307116391657027667, 6.34414636142208508345065938004, 8.767067297803237727133546199533, 9.842588591796313770506766876314, 10.58105813501574886387679093937, 11.42110596998024112853202092295, 12.97456034717401900210201053496