L(s) = 1 | − 1.46·2-s + (1.16 − 1.28i)3-s + 0.139·4-s + 0.594i·5-s + (−1.69 + 1.87i)6-s + 1.86·7-s + 2.72·8-s + (−0.300 − 2.98i)9-s − 0.869i·10-s + 0.676·11-s + (0.161 − 0.178i)12-s − 5.37i·13-s − 2.72·14-s + (0.763 + 0.690i)15-s − 4.25·16-s − 1.70i·17-s + ⋯ |
L(s) = 1 | − 1.03·2-s + (0.670 − 0.741i)3-s + 0.0695·4-s + 0.265i·5-s + (−0.693 + 0.767i)6-s + 0.703·7-s + 0.962·8-s + (−0.100 − 0.994i)9-s − 0.274i·10-s + 0.204·11-s + (0.0466 − 0.0516i)12-s − 1.49i·13-s − 0.727·14-s + (0.197 + 0.178i)15-s − 1.06·16-s − 0.412i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.746991 - 0.403201i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.746991 - 0.403201i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.16 + 1.28i)T \) |
| 59 | \( 1 + (-1.93 + 7.43i)T \) |
good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 5 | \( 1 - 0.594iT - 5T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 - 0.676T + 11T^{2} \) |
| 13 | \( 1 + 5.37iT - 13T^{2} \) |
| 17 | \( 1 + 1.70iT - 17T^{2} \) |
| 19 | \( 1 + 2.25T + 19T^{2} \) |
| 23 | \( 1 - 4.86T + 23T^{2} \) |
| 29 | \( 1 - 6.24iT - 29T^{2} \) |
| 31 | \( 1 + 2.48iT - 31T^{2} \) |
| 37 | \( 1 - 7.86iT - 37T^{2} \) |
| 41 | \( 1 - 2.29iT - 41T^{2} \) |
| 43 | \( 1 - 7.11iT - 43T^{2} \) |
| 47 | \( 1 + 8.46T + 47T^{2} \) |
| 53 | \( 1 - 5.73iT - 53T^{2} \) |
| 61 | \( 1 - 10.0iT - 61T^{2} \) |
| 67 | \( 1 - 5.37iT - 67T^{2} \) |
| 71 | \( 1 + 5.92iT - 71T^{2} \) |
| 73 | \( 1 + 8.26iT - 73T^{2} \) |
| 79 | \( 1 - 8.29T + 79T^{2} \) |
| 83 | \( 1 - 0.0941T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 17.8iT - 97T^{2} \) |
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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
−12.71901281549934940078551221199, −11.32134865304277437693689637908, −10.40572089071951141356547765419, −9.283157073546609192752580622838, −8.375632114285907654383671805369, −7.74723413978053675332335987773, −6.69370075700139909522987664644, −4.91737230872597404630533303827, −3.01009819221546976033602332409, −1.22245333704650313101855834760,
1.90169836458706514322098136145, 4.06977282938096059805462890001, 4.96340666105419233682169571608, 6.99717277657502639984462377315, 8.220486455851872928995462600791, 8.860679386919625725725583000255, 9.566657317995193996788234351183, 10.65401409469725502124176784861, 11.41174452074998572717206349093, 12.97160273906311844282082172769