L(s) = 1 | − 297. i·2-s + 5.03e3i·3-s − 5.56e4·4-s − 1.22e5·5-s + 1.49e6·6-s + 2.97e6·7-s + 6.80e6i·8-s − 1.10e7·9-s + 3.63e7i·10-s + 6.17e6i·11-s − 2.80e8i·12-s − 4.98e7·13-s − 8.83e8i·14-s − 6.15e8i·15-s + 2.00e8·16-s − 2.30e9i·17-s + ⋯ |
L(s) = 1 | − 1.64i·2-s + 1.32i·3-s − 1.69·4-s − 0.699·5-s + 2.18·6-s + 1.36·7-s + 1.14i·8-s − 0.766·9-s + 1.14i·10-s + 0.0954i·11-s − 2.25i·12-s − 0.220·13-s − 2.23i·14-s − 0.929i·15-s + 0.186·16-s − 1.36i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(8)\) |
\(\approx\) |
\(0.8139536054\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8139536054\) |
\(L(\frac{17}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (-2.92e10 + 8.81e10i)T \) |
good | 2 | \( 1 + 297. iT - 3.27e4T^{2} \) |
| 3 | \( 1 - 5.03e3iT - 1.43e7T^{2} \) |
| 5 | \( 1 + 1.22e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 2.97e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 6.17e6iT - 4.17e15T^{2} \) |
| 13 | \( 1 + 4.98e7T + 5.11e16T^{2} \) |
| 17 | \( 1 + 2.30e9iT - 2.86e18T^{2} \) |
| 19 | \( 1 - 3.73e9iT - 1.51e19T^{2} \) |
| 23 | \( 1 + 2.07e9T + 2.66e20T^{2} \) |
| 31 | \( 1 - 1.24e11iT - 2.34e22T^{2} \) |
| 37 | \( 1 - 7.98e11iT - 3.33e23T^{2} \) |
| 41 | \( 1 - 7.89e11iT - 1.55e24T^{2} \) |
| 43 | \( 1 - 1.01e12iT - 3.17e24T^{2} \) |
| 47 | \( 1 - 1.01e12iT - 1.20e25T^{2} \) |
| 53 | \( 1 + 6.28e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 3.10e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 2.04e13iT - 6.02e26T^{2} \) |
| 67 | \( 1 - 1.18e12T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.12e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 5.76e13iT - 8.90e27T^{2} \) |
| 79 | \( 1 + 9.00e13iT - 2.91e28T^{2} \) |
| 83 | \( 1 + 1.65e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 8.14e14iT - 1.74e29T^{2} \) |
| 97 | \( 1 - 1.08e15iT - 6.33e29T^{2} \) |
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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
−13.86238559612652539374077484623, −12.01644331180363423481093852855, −11.39336580587408539522433715327, −10.38026160511409456414017271304, −9.402782135607141364804709829038, −7.997170274379816679688465672781, −4.88961133383398656771255543503, −4.24547515302345766714022457138, −2.99168066581261082420451069178, −1.39438395201790389868887884930,
0.24018345214340489045207428010, 1.79507140251367049610924879796, 4.40054284971947537096268444120, 5.80873903871223830853841824081, 7.12181023462375346962017511358, 7.83413621174736903584028877488, 8.630848832116637286840585545090, 11.20666481387061300742663309129, 12.54100103886354981496505369295, 13.79861137472724909428414623660