L(s) = 1 | + (1.45 + 0.389i)2-s + (0.239 − 1.71i)3-s + (0.232 + 0.133i)4-s + (−1.06 + 1.06i)5-s + (1.01 − 2.40i)6-s + (−0.366 − 1.36i)7-s + (−1.84 − 1.84i)8-s + (−2.88 − 0.820i)9-s + (−1.96 + 1.13i)10-s + (1.06 − 3.97i)11-s + (0.285 − 0.366i)12-s − 2.12i·14-s + (1.57 + 2.08i)15-s + (−2.23 − 3.86i)16-s + (2.51 − 4.36i)17-s + (−3.87 − 2.31i)18-s + ⋯ |
L(s) = 1 | + (1.02 + 0.275i)2-s + (0.138 − 0.990i)3-s + (0.116 + 0.0669i)4-s + (−0.476 + 0.476i)5-s + (0.415 − 0.980i)6-s + (−0.138 − 0.516i)7-s + (−0.652 − 0.652i)8-s + (−0.961 − 0.273i)9-s + (−0.621 + 0.358i)10-s + (0.321 − 1.19i)11-s + (0.0823 − 0.105i)12-s − 0.569i·14-s + (0.405 + 0.537i)15-s + (−0.558 − 0.966i)16-s + (0.611 − 1.05i)17-s + (−0.913 − 0.546i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17635 - 1.38750i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17635 - 1.38750i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.239 + 1.71i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.45 - 0.389i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (1.06 - 1.06i)T - 5iT^{2} \) |
| 7 | \( 1 + (0.366 + 1.36i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.06 + 3.97i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.51 + 4.36i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.73 + i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (6.20 - 3.58i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.46 - 2.46i)T + 31iT^{2} \) |
| 37 | \( 1 + (-5.23 - 1.40i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (5.42 + 1.45i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.90 - 1.09i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.25 - 4.25i)T + 47iT^{2} \) |
| 53 | \( 1 + 0.779iT - 53T^{2} \) |
| 59 | \( 1 + (-2.90 + 0.779i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.53 + 5.73i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.779 - 2.90i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.901 + 0.901i)T - 73iT^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + (2.90 - 2.90i)T - 83iT^{2} \) |
| 89 | \( 1 + (-2.41 + 9.01i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (1.63 - 0.437i)T + (84.0 - 48.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.13459446361007619147815593747, −9.681796048600629115000054129884, −8.759370404679563521108433437911, −7.53534060480870806986633339642, −6.97780398592058894615630820970, −5.99603475215007394454943284758, −5.15454028055749951183012365732, −3.61061634248376991698046208201, −3.05290954055775619305096693903, −0.796253900134437259265293530987,
2.40366690777779871382732690868, 3.73030492021070528036463167990, 4.27807264469431567537230801161, 5.25125354013980950394506854291, 6.02450263411121252227721323084, 7.72383302577349059036198348169, 8.612381137502497429683159184975, 9.449909728911117323407350725741, 10.23966073788772812155003696440, 11.50536823571280956214810108644