L(s) = 1 | + (10.7 − 11.3i)3-s − 4.25·5-s + 187. i·7-s + (−12.5 − 242. i)9-s + 712. i·11-s − 804. i·13-s + (−45.6 + 48.0i)15-s + 309. i·17-s − 2.46e3·19-s + (2.11e3 + 2.01e3i)21-s + 3.06e3·23-s − 3.10e3·25-s + (−2.87e3 − 2.46e3i)27-s − 6.32e3·29-s − 9.00e3i·31-s + ⋯ |
L(s) = 1 | + (0.688 − 0.725i)3-s − 0.0760·5-s + 1.44i·7-s + (−0.0516 − 0.998i)9-s + 1.77i·11-s − 1.31i·13-s + (−0.0523 + 0.0551i)15-s + 0.259i·17-s − 1.56·19-s + (1.04 + 0.995i)21-s + 1.20·23-s − 0.994·25-s + (−0.759 − 0.650i)27-s − 1.39·29-s − 1.68i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0258i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.2325162939\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2325162939\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-10.7 + 11.3i)T \) |
good | 5 | \( 1 + 4.25T + 3.12e3T^{2} \) |
| 7 | \( 1 - 187. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 712. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 804. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 309. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.46e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.00e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 7.35e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.03e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 3.83e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.39e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.68e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.54e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 3.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.74e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.10e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 2.07e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.36e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 9.40e4T + 8.58e9T^{2} \) |
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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
−9.771889885579472926112310410315, −9.111582116645439363890187046482, −8.141844559668404201233658033148, −7.42073656861734380488771991476, −6.28558899882498518267867080308, −5.32172659408925455778191451703, −3.88547877652131510130049613359, −2.45849324285858391785491451847, −1.92053547690892889303687511294, −0.04774305056631611499042546569,
1.51499351829419733402638301527, 3.14870167320663463553056692828, 3.94283365849529389847581248748, 4.82535736714282055840104605913, 6.31981965936684960811969490248, 7.32360977209393399896496304558, 8.405828012841188096630085835259, 9.038161956905214996988714212026, 10.11405482981332492221280941165, 10.95687705279468845429120231320