L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (−2 − 1.73i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 13-s − 0.999·15-s + (−1.5 − 2.59i)19-s + (2.5 − 0.866i)21-s + (3.5 + 6.06i)23-s + (−0.499 + 0.866i)25-s + 0.999·27-s − 8·29-s + (−1 + 1.73i)31-s + (−0.499 − 0.866i)33-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (−0.755 − 0.654i)7-s + (−0.166 − 0.288i)9-s + (−0.150 + 0.261i)11-s + 0.277·13-s − 0.258·15-s + (−0.344 − 0.596i)19-s + (0.545 − 0.188i)21-s + (0.729 + 1.26i)23-s + (−0.0999 + 0.173i)25-s + 0.192·27-s − 1.48·29-s + (−0.179 + 0.311i)31-s + (−0.0870 − 0.150i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2200321770\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2200321770\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 11T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (2.5 + 4.33i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.5 + 9.52i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−9.309689410272044474232605311289, −8.363259098663911736200420303808, −7.08918743127239478044254391619, −6.89001979491523585099234649437, −5.70158173067212268174121192934, −5.04168140523126230106896089692, −3.78222559471431756922802333872, −3.30921537606468101755556365988, −1.84697273236623692140284270232, −0.084000489316477839434140689195,
1.49137808468798994103135524713, 2.63533721770063924908287739903, 3.64969016491551752799892052416, 4.92545889399787184421972231256, 5.67792637778453247867409500125, 6.41807598244263102116197279392, 7.07423613665469414586408719283, 8.312083123535375178383890198474, 8.666328106996505670956017161449, 9.645529194206775674237179521887