L(s) = 1 | + (0.663 − 1.14i)2-s + (3.11 + 5.40i)4-s + (−2.5 − 4.33i)5-s + (12.0 − 20.8i)7-s + 18.8·8-s − 6.63·10-s + (−4.13 + 7.16i)11-s + (−43.5 − 75.4i)13-s + (−15.9 − 27.6i)14-s + (−12.4 + 21.5i)16-s + 51.9·17-s − 88.5·19-s + (15.5 − 27.0i)20-s + (5.49 + 9.50i)22-s + (64.6 + 111. i)23-s + ⋯ |
L(s) = 1 | + (0.234 − 0.406i)2-s + (0.389 + 0.675i)4-s + (−0.223 − 0.387i)5-s + (0.650 − 1.12i)7-s + 0.834·8-s − 0.209·10-s + (−0.113 + 0.196i)11-s + (−0.929 − 1.60i)13-s + (−0.305 − 0.528i)14-s + (−0.194 + 0.336i)16-s + 0.740·17-s − 1.06·19-s + (0.174 − 0.302i)20-s + (0.0532 + 0.0921i)22-s + (0.585 + 1.01i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.159184494\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.159184494\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.5 + 4.33i)T \) |
good | 2 | \( 1 + (-0.663 + 1.14i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-12.0 + 20.8i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (4.13 - 7.16i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (43.5 + 75.4i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 51.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-64.6 - 111. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-135. + 234. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (112. + 194. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 70.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + (183. + 317. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-97.7 + 169. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-179. + 311. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 29.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-429. - 743. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-278. + 482. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-20.9 - 36.2i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 549.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 185.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (40.2 - 69.7i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-288. + 499. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 224.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-277. + 480. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.57923249199967048948263546958, −10.07507764983001131432173331206, −8.484438266742376500738237860192, −7.65544066549582945553408967698, −7.25050352930208927182192945129, −5.52911121798089160820267471142, −4.44106294886030842788277060920, −3.55731821870574611346683560993, −2.22243307669330381926379856626, −0.65705203914212283968849517849,
1.62662391572516285998085203965, 2.69442909306504281969623421334, 4.56083443496268524887982704652, 5.26336592818974192484330983298, 6.46323701174723611779175564451, 7.05359272432462352191034594414, 8.325192402265418749570829564193, 9.176285225920292328634171524119, 10.31656785401115664316288099327, 11.10039648964909902595172570246