L(s) = 1 | + (−1.02 + 1.98i)5-s + (1.10 + 0.640i)7-s + (2.07 − 3.58i)11-s + (−5.64 + 3.26i)13-s + 5.98i·17-s − 7.17·19-s + (6.52 − 3.76i)23-s + (−2.88 − 4.08i)25-s + (−2.59 + 4.5i)29-s + (−2.58 − 4.48i)31-s + (−2.41 + 1.54i)35-s + 5.24i·37-s + (0.340 + 0.589i)41-s + (−1.10 − 0.640i)43-s + (−4.59 − 2.65i)47-s + ⋯ |
L(s) = 1 | + (−0.459 + 0.888i)5-s + (0.419 + 0.242i)7-s + (0.624 − 1.08i)11-s + (−1.56 + 0.904i)13-s + 1.45i·17-s − 1.64·19-s + (1.35 − 0.785i)23-s + (−0.577 − 0.816i)25-s + (−0.482 + 0.835i)29-s + (−0.465 − 0.805i)31-s + (−0.407 + 0.261i)35-s + 0.861i·37-s + (0.0531 + 0.0920i)41-s + (−0.169 − 0.0977i)43-s + (−0.670 − 0.387i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5087867581\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5087867581\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.02 - 1.98i)T \) |
good | 7 | \( 1 + (-1.10 - 0.640i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.07 + 3.58i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.64 - 3.26i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 5.98iT - 17T^{2} \) |
| 19 | \( 1 + 7.17T + 19T^{2} \) |
| 23 | \( 1 + (-6.52 + 3.76i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.59 - 4.5i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.58 + 4.48i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.24iT - 37T^{2} \) |
| 41 | \( 1 + (-0.340 - 0.589i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.10 + 0.640i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.59 + 2.65i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 2.21iT - 53T^{2} \) |
| 59 | \( 1 + (-3.80 - 6.58i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.08 + 1.88i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (13.5 - 7.80i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.50T + 71T^{2} \) |
| 73 | \( 1 - 7.80iT - 73T^{2} \) |
| 79 | \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (8.44 + 4.87i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + (12.4 + 7.16i)T + (48.5 + 84.0i)T^{2} \) |
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
−9.855307927559537301466601072519, −8.707488174830474916590813446129, −8.406191094524772189348751684933, −7.21561526384735069951599908967, −6.66967189742643270481125593233, −5.87455670931862221716612769530, −4.64871277504302271874226934272, −3.91356717912131544746524858746, −2.83662662492730161511520446093, −1.82315386146693873733960519861,
0.18676377983837126793547797944, 1.64315456442881756756998308671, 2.84508399626775019464496871422, 4.21196702228372324554840726254, 4.80337659565285973432206119591, 5.42695153070293314552353413716, 6.89298763950229145037237599190, 7.43478349817242254682685027980, 8.104110737769747212856677544382, 9.266837188404603513026628813663