L(s) = 1 | + (−1.31 + 0.523i)2-s − 3-s + (1.45 − 1.37i)4-s − 1.71i·5-s + (1.31 − 0.523i)6-s − 3.43·7-s + (−1.18 + 2.56i)8-s + 9-s + (0.898 + 2.25i)10-s + 1.17·11-s + (−1.45 + 1.37i)12-s − 1.11i·13-s + (4.51 − 1.79i)14-s + 1.71i·15-s + (0.216 − 3.99i)16-s − 6.00·17-s + ⋯ |
L(s) = 1 | + (−0.928 + 0.370i)2-s − 0.577·3-s + (0.725 − 0.687i)4-s − 0.767i·5-s + (0.536 − 0.213i)6-s − 1.29·7-s + (−0.419 + 0.907i)8-s + 0.333·9-s + (0.284 + 0.713i)10-s + 0.352·11-s + (−0.419 + 0.397i)12-s − 0.308i·13-s + (1.20 − 0.480i)14-s + 0.443i·15-s + (0.0540 − 0.998i)16-s − 1.45·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.185378 + 0.253882i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.185378 + 0.253882i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.31 - 0.523i)T \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + (3.55 - 7.37i)T \) |
good | 5 | \( 1 + 1.71iT - 5T^{2} \) |
| 7 | \( 1 + 3.43T + 7T^{2} \) |
| 11 | \( 1 - 1.17T + 11T^{2} \) |
| 13 | \( 1 + 1.11iT - 13T^{2} \) |
| 17 | \( 1 + 6.00T + 17T^{2} \) |
| 19 | \( 1 + 1.61iT - 19T^{2} \) |
| 23 | \( 1 - 5.58iT - 23T^{2} \) |
| 29 | \( 1 + 2.67T + 29T^{2} \) |
| 31 | \( 1 - 3.52T + 31T^{2} \) |
| 37 | \( 1 - 4.31T + 37T^{2} \) |
| 41 | \( 1 - 8.77iT - 41T^{2} \) |
| 43 | \( 1 - 1.55T + 43T^{2} \) |
| 47 | \( 1 - 3.12iT - 47T^{2} \) |
| 53 | \( 1 - 9.63iT - 53T^{2} \) |
| 59 | \( 1 + 5.48iT - 59T^{2} \) |
| 61 | \( 1 - 3.17iT - 61T^{2} \) |
| 71 | \( 1 - 14.4iT - 71T^{2} \) |
| 73 | \( 1 + 7.90T + 73T^{2} \) |
| 79 | \( 1 - 7.48T + 79T^{2} \) |
| 83 | \( 1 + 3.57iT - 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 7.37iT - 97T^{2} \) |
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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
−10.34531262532359272580054406063, −9.404878864719264685910792771111, −9.119489810502045194885039123892, −8.007794996080641825864169238727, −6.94838959332406179423949501932, −6.34007973290312143878694925908, −5.48770484436373956539020977197, −4.35113867105677662016021713251, −2.77674510467960109352593040758, −1.12925062479673683430958988037,
0.25679119546566664696349043941, 2.17981309918463782942555863308, 3.24046882268778155194509097937, 4.30957703117386824142388136008, 6.11141769480818798395275720407, 6.64714339590753237520145075045, 7.20104640981079284127331474292, 8.538604514379387144971427405708, 9.306267979911623786379830362689, 10.09368703764850024040034226859