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
−14.21849421328805753949467842849, −12.99581120630903484402693634626, −12.36057736510634857375458834021, −10.94532583001237446605104749711, −9.706824278428610577373155902898, −8.250019655275872751043527251637, −7.08585544143943675140507218754, −5.57709364750440716429837695488, −5.00212064571773703624658333110, −2.99109732830900938047165688224,
2.44875023367782687263432536837, 4.24605824971914652230569860549, 4.96092440515681747857783805520, 7.03175841904256586093581601287, 8.077741216665055473900571049257, 9.796024811089212381660377861908, 10.85663087061310372691406761102, 11.84797977893248488195440637564, 12.72604380439538797615311341720, 13.72142918954134949324861283565