L(s) = 1 | + (0.866 − 0.5i)2-s + (0.866 − 1.5i)3-s + (−1.5 + 2.59i)4-s − 1.73i·6-s + (4.33 + 5.5i)7-s + 7i·8-s + (−1.5 − 2.59i)9-s + (−2 + 3.46i)11-s + (2.59 + 4.5i)12-s − 17.3·13-s + (6.5 + 2.59i)14-s + (−2.5 − 4.33i)16-s + (−4.33 + 7.5i)17-s + (−2.59 − 1.5i)18-s + (−6 + 3.46i)19-s + ⋯ |
L(s) = 1 | + (0.433 − 0.250i)2-s + (0.288 − 0.5i)3-s + (−0.375 + 0.649i)4-s − 0.288i·6-s + (0.618 + 0.785i)7-s + 0.875i·8-s + (−0.166 − 0.288i)9-s + (−0.181 + 0.314i)11-s + (0.216 + 0.375i)12-s − 1.33·13-s + (0.464 + 0.185i)14-s + (−0.156 − 0.270i)16-s + (−0.254 + 0.441i)17-s + (−0.144 − 0.0833i)18-s + (−0.315 + 0.182i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.439916031\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.439916031\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-4.33 - 5.5i)T \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T + (2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 17.3T + 169T^{2} \) |
| 17 | \( 1 + (4.33 - 7.5i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (6 - 3.46i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (26.8 - 15.5i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 10T + 841T^{2} \) |
| 31 | \( 1 + (-25.5 - 14.7i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-43.3 + 25i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 53.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 34iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-21.6 - 37.5i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (43.3 + 25i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-48 - 27.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-21 + 12.1i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (43.3 + 25i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 97T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-24.2 + 42i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (3.5 + 6.06i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 152.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-136.5 + 78.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 112.T + 9.40e3T^{2} \) |
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
−11.28071013252010929486646266983, −9.888033025971341832191718823016, −9.041656468633495455351568709152, −8.000522196084031955734182633482, −7.67292244267177706821283808339, −6.22356117295626314533666594047, −5.09287692245592238230875402589, −4.23425339061767256269062236149, −2.83114775680661672339498630254, −1.99046080860034699279562930730,
0.45245919752923923337974821011, 2.30580068868431604055166213844, 3.96468684315909682896835975520, 4.64322326188278690484952797707, 5.47749040742889504634370423470, 6.68029885688259014762340103595, 7.66669325266151320348869810832, 8.651057387043193042913171818669, 9.754770388400180654975240067382, 10.21831035924805662000235643896