L(s) = 1 | + (1.30 − 0.951i)2-s + (0.190 − 0.587i)4-s + (−0.309 − 0.224i)5-s + (0.927 − 2.85i)7-s + (0.690 + 2.12i)8-s − 0.618·10-s + (−2.80 + 1.76i)11-s + (−5.04 + 3.66i)13-s + (−1.5 − 4.61i)14-s + (3.92 + 2.85i)16-s + (−0.5 − 0.363i)17-s + (−0.263 − 0.812i)19-s + (−0.190 + 0.138i)20-s + (−2 + 4.97i)22-s + 5.47·23-s + ⋯ |
L(s) = 1 | + (0.925 − 0.672i)2-s + (0.0954 − 0.293i)4-s + (−0.138 − 0.100i)5-s + (0.350 − 1.07i)7-s + (0.244 + 0.751i)8-s − 0.195·10-s + (−0.846 + 0.531i)11-s + (−1.39 + 1.01i)13-s + (−0.400 − 1.23i)14-s + (0.981 + 0.713i)16-s + (−0.121 − 0.0881i)17-s + (−0.0605 − 0.186i)19-s + (−0.0427 + 0.0310i)20-s + (−0.426 + 1.06i)22-s + 1.14·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36063 - 0.512020i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36063 - 0.512020i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (2.80 - 1.76i)T \) |
good | 2 | \( 1 + (-1.30 + 0.951i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (0.309 + 0.224i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.927 + 2.85i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (5.04 - 3.66i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.363i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.263 + 0.812i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 + (-1.38 + 4.25i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.11 + 2.26i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.30 - 4.02i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.83 + 5.65i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 + (-0.190 - 0.587i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (5.97 - 4.33i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.64 + 5.06i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.927 + 0.673i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + (-11.7 - 8.55i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.381 + 1.17i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.427 - 0.310i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (10.2 + 7.46i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 9.47T + 89T^{2} \) |
| 97 | \( 1 + (12.1 - 8.83i)T + (29.9 - 92.2i)T^{2} \) |
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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
−13.72142918954134949324861283565, −12.72604380439538797615311341720, −11.84797977893248488195440637564, −10.85663087061310372691406761102, −9.796024811089212381660377861908, −8.077741216665055473900571049257, −7.03175841904256586093581601287, −4.96092440515681747857783805520, −4.24605824971914652230569860549, −2.44875023367782687263432536837,
2.99109732830900938047165688224, 5.00212064571773703624658333110, 5.57709364750440716429837695488, 7.08585544143943675140507218754, 8.250019655275872751043527251637, 9.706824278428610577373155902898, 10.94532583001237446605104749711, 12.36057736510634857375458834021, 12.99581120630903484402693634626, 14.21849421328805753949467842849