L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.5 + 2.59i)5-s + 7-s + 0.999·8-s + (−1.5 − 2.59i)10-s + 6·11-s + (−2.5 − 4.33i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−3 + 5.19i)17-s + (−0.5 + 4.33i)19-s + 3·20-s + (−3 + 5.19i)22-s + (−1.5 − 2.59i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.670 + 1.16i)5-s + 0.377·7-s + 0.353·8-s + (−0.474 − 0.821i)10-s + 1.80·11-s + (−0.693 − 1.20i)13-s + (−0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.727 + 1.26i)17-s + (−0.114 + 0.993i)19-s + 0.670·20-s + (−0.639 + 1.10i)22-s + (−0.312 − 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.066065173\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.066065173\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + (0.5 - 4.33i)T \) |
good | 5 | \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.5 - 2.59i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.065089087978926621678223299756, −8.478964609417662127624443359851, −7.59466234094809917145286280806, −7.14586737733098363587534372286, −6.26617674338902892488988018934, −5.74560766387340975041462990873, −4.30678440294243884799756986519, −3.83647788796639214552358228690, −2.64430404921242313233186325293, −1.29042999942740949366716676356,
0.45862839633879342325185375475, 1.49794307760171434164442726551, 2.56310454661527329649042056031, 4.08514119385663899794742989753, 4.32033307344482117162599873241, 5.12291520735608662543774538075, 6.53978376501671198090063981037, 7.17115200286137465120695092347, 8.088845259334253567837359767743, 8.877290747695455478508886466250