L(s) = 1 | + (−0.917 + 1.58i)2-s + (1.09 + 1.90i)3-s + (−0.682 − 1.18i)4-s + (0.317 − 0.550i)5-s − 4.03·6-s + (0.317 − 2.62i)7-s − 1.16·8-s + (−0.917 + 1.58i)9-s + (0.582 + 1.00i)10-s + (0.5 + 0.866i)11-s + (1.50 − 2.59i)12-s − 1.80·13-s + (3.88 + 2.91i)14-s + 1.39·15-s + (2.43 − 4.21i)16-s + (1.41 + 2.45i)17-s + ⋯ |
L(s) = 1 | + (−0.648 + 1.12i)2-s + (0.634 + 1.09i)3-s + (−0.341 − 0.590i)4-s + (0.142 − 0.246i)5-s − 1.64·6-s + (0.120 − 0.992i)7-s − 0.412·8-s + (−0.305 + 0.529i)9-s + (0.184 + 0.319i)10-s + (0.150 + 0.261i)11-s + (0.433 − 0.750i)12-s − 0.499·13-s + (1.03 + 0.778i)14-s + 0.360·15-s + (0.608 − 1.05i)16-s + (0.343 + 0.595i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.438450 + 0.711756i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.438450 + 0.711756i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.317 + 2.62i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.917 - 1.58i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.09 - 1.90i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.317 + 0.550i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 + (-1.41 - 2.45i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.78 + 4.81i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.08 - 1.87i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + (3.21 + 5.56i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.03 - 5.25i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 7.53T + 41T^{2} \) |
| 43 | \( 1 + 4.86T + 43T^{2} \) |
| 47 | \( 1 + (-1.41 + 2.45i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.73 + 6.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.90 + 10.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.16 + 3.75i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.801 - 1.38i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.29T + 71T^{2} \) |
| 73 | \( 1 + (-7.99 - 13.8i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.38 - 4.12i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.23T + 83T^{2} \) |
| 89 | \( 1 + (0.182 - 0.315i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.59T + 97T^{2} \) |
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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
−15.03987601597283250393601055935, −14.33975489412468285280248672387, −12.98382634995223429557658725931, −11.21446965131462365671985978180, −9.771689068429543228830677506353, −9.288134338527491058066729846521, −7.967490264424783029488090657076, −6.97300884166097984311781674530, −5.18601066721870050876048389686, −3.63727125864432032688310576714,
1.82956938649959526708922619529, 3.00018392625285238886447387149, 5.84755281498972572103272440736, 7.46548999128718578251289408523, 8.645393992514118502655631442366, 9.576757156818804024253135938294, 10.88909061010808921347956076822, 12.15282026018480184890961102392, 12.55918558264795160720557839665, 14.04052957157739843485601002014