L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−1.23 − 1.86i)5-s − 0.999·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.133 − 2.23i)10-s + (−0.866 − 0.499i)12-s + 2i·13-s + (2 + i)15-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + (0.866 − 0.499i)18-s + (1 − 1.73i)19-s + (1 − 1.99i)20-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.550 − 0.834i)5-s − 0.408·6-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.0423 − 0.705i)10-s + (−0.249 − 0.144i)12-s + 0.554i·13-s + (0.516 + 0.258i)15-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + (0.204 − 0.117i)18-s + (0.229 − 0.397i)19-s + (0.223 − 0.447i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.789239948\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.789239948\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (1.23 + 1.86i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.92 - 4i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.19 - 3i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + (-5.19 - 3i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (12.1 - 7i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-8.66 + 5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.364644738357834676154701354737, −8.932483548337326150243801926880, −7.87474438767230214558840767929, −7.15243733938815151775279202086, −6.23820042770512513241332509429, −5.36293973070893895827501179779, −4.58911067733755502382119721115, −4.02646727411456748207857783367, −2.78782435644693789294669359927, −1.07840116367716814702412847862,
0.78514812779563027885484686310, 2.43369053393945337683509183049, 3.21636513982893502262985115531, 4.31908122275157766177104608443, 5.13266709729022515351766736213, 6.16780896932967141824504476078, 6.80034222082921771865042816481, 7.55444412196399884711211372646, 8.465462099071601549959134151427, 9.613432398855818275223047347488