L(s) = 1 | + 2.12i·2-s + (−2.98 − 0.332i)3-s − 0.528·4-s − 5.15i·5-s + (0.707 − 6.34i)6-s + 2.69·7-s + 7.38i·8-s + (8.77 + 1.98i)9-s + 10.9·10-s + 15.4i·11-s + (1.57 + 0.175i)12-s + 18.6·13-s + 5.72i·14-s + (−1.71 + 15.3i)15-s − 17.8·16-s − 4.88i·17-s + ⋯ |
L(s) = 1 | + 1.06i·2-s + (−0.993 − 0.110i)3-s − 0.132·4-s − 1.03i·5-s + (0.117 − 1.05i)6-s + 0.384·7-s + 0.923i·8-s + (0.975 + 0.220i)9-s + 1.09·10-s + 1.40i·11-s + (0.131 + 0.0146i)12-s + 1.43·13-s + 0.409i·14-s + (−0.114 + 1.02i)15-s − 1.11·16-s − 0.287i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.966943 + 0.865053i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.966943 + 0.865053i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.98 + 0.332i)T \) |
| 59 | \( 1 + 7.68iT \) |
good | 2 | \( 1 - 2.12iT - 4T^{2} \) |
| 5 | \( 1 + 5.15iT - 25T^{2} \) |
| 7 | \( 1 - 2.69T + 49T^{2} \) |
| 11 | \( 1 - 15.4iT - 121T^{2} \) |
| 13 | \( 1 - 18.6T + 169T^{2} \) |
| 17 | \( 1 + 4.88iT - 289T^{2} \) |
| 19 | \( 1 - 3.41T + 361T^{2} \) |
| 23 | \( 1 - 15.9iT - 529T^{2} \) |
| 29 | \( 1 - 24.8iT - 841T^{2} \) |
| 31 | \( 1 - 28.0T + 961T^{2} \) |
| 37 | \( 1 - 60.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 37.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 15.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 13.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 35.0iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 83.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 72.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 27.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 54.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 123.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 143. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 21.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 98.9T + 9.40e3T^{2} \) |
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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
−12.67400951976303250703823591907, −11.74918604676787927215440354284, −10.89946691764108517844063542112, −9.525776775863805581641718098278, −8.330073281218276815081411989297, −7.35240087181283330436015579874, −6.33811035358476882402620649273, −5.28975411688002147462757139372, −4.50574855630646126701970525128, −1.48625283974917872200328538798,
1.05765986099746576979933698929, 2.95745887874184649452005842311, 4.13841856234141466642267226766, 6.01464591315058896903286780501, 6.60380316749491153935177332743, 8.186194639712562349187513283527, 9.753629432463632265057837857571, 10.83117989363319274178805199336, 11.06232722815185548559673965719, 11.76096659689974167702293032995