L(s) = 1 | − 4.54i·2-s + (2.93 − 4.28i)3-s − 12.6·4-s − 11.6·5-s + (−19.4 − 13.3i)6-s + 21.1i·8-s + (−9.72 − 25.1i)9-s + 52.7i·10-s − 17.9i·11-s + (−37.2 + 54.2i)12-s + 62.4i·13-s + (−34.1 + 49.7i)15-s − 4.97·16-s + 21.4·17-s + (−114. + 44.1i)18-s − 10.9i·19-s + ⋯ |
L(s) = 1 | − 1.60i·2-s + (0.565 − 0.824i)3-s − 1.58·4-s − 1.03·5-s + (−1.32 − 0.909i)6-s + 0.936i·8-s + (−0.360 − 0.932i)9-s + 1.66i·10-s − 0.491i·11-s + (−0.895 + 1.30i)12-s + 1.33i·13-s + (−0.587 + 0.855i)15-s − 0.0777·16-s + 0.305·17-s + (−1.49 + 0.578i)18-s − 0.132i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.624761 + 0.699294i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.624761 + 0.699294i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.93 + 4.28i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 4.54iT - 8T^{2} \) |
| 5 | \( 1 + 11.6T + 125T^{2} \) |
| 11 | \( 1 + 17.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 62.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 21.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 10.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 69.0iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 265. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 10.2iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 41.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 31.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 224.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 163.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 527. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 411.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 258. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 323.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 45.4iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 562. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 289.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 448.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 561.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 214. iT - 9.12e5T^{2} \) |
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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
−11.68807995215149371682035264428, −11.38132208091442766782411213737, −9.868050640338275001804760565111, −8.849922706590220429555954702969, −7.898705563252926836667860081758, −6.56095679665319548315243427630, −4.31847883148021139753607808685, −3.33137254216542675239001762537, −1.99204651913730713804173766143, −0.42576483022464004482419783046,
3.36559498331477967299174032619, 4.63865493663607330155469255830, 5.62375830095227687252299993367, 7.28598424772383993308120425194, 7.941258231586094200519233450647, 8.770600821417188795038406821239, 9.956539385834118507054819284762, 11.14819884462100450862129652029, 12.59844703167511382982006276865, 13.78164357806303729583224017605