L(s) = 1 | + (−1 − 1.41i)3-s + (−0.5 + 0.866i)5-s + (0.724 + 1.25i)7-s + (−1.00 + 2.82i)9-s + (−1.72 − 2.98i)11-s + (−1.94 + 3.37i)13-s + (1.72 − 0.158i)15-s − 4.89·17-s − 4·19-s + (1.05 − 2.28i)21-s + (0.275 − 0.476i)23-s + (2 + 3.46i)25-s + (5.00 − 1.41i)27-s + (4.94 + 8.57i)29-s + (−3.72 + 6.45i)31-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.816i)3-s + (−0.223 + 0.387i)5-s + (0.273 + 0.474i)7-s + (−0.333 + 0.942i)9-s + (−0.520 − 0.900i)11-s + (−0.540 + 0.936i)13-s + (0.445 − 0.0410i)15-s − 1.18·17-s − 0.917·19-s + (0.229 − 0.497i)21-s + (0.0573 − 0.0994i)23-s + (0.400 + 0.692i)25-s + (0.962 − 0.272i)27-s + (0.919 + 1.59i)29-s + (−0.668 + 1.15i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.268005 + 0.386535i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.268005 + 0.386535i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.724 - 1.25i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.72 + 2.98i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.94 - 3.37i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-0.275 + 0.476i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.94 - 8.57i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.72 - 6.45i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.89T + 37T^{2} \) |
| 41 | \( 1 + (-1.05 + 1.81i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.17 - 10.6i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.17 + 7.22i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 0.898T + 53T^{2} \) |
| 59 | \( 1 + (0.174 - 0.301i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.949 - 1.64i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.17 - 2.03i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 + (4.27 + 7.40i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.72 + 4.71i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 + (2.94 + 5.10i)T + (-48.5 + 84.0i)T^{2} \) |
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
−10.97813553370871998394931979290, −10.56856485131131733105363901435, −8.898705811167742626945685369476, −8.451853873308789095398499946454, −7.11732569612177249784188057910, −6.69501396869844011243695972227, −5.53548095538216404492859524390, −4.65432499601786570478946862812, −2.98594603016891087920868813104, −1.79590261645023774044436844935,
0.27164024523269784222340370229, 2.47293557828317056910520970069, 4.16021212456707605471521812425, 4.62983801971823035040113728822, 5.67792344196394341805743545864, 6.79196902073354319482678750417, 7.84384654109947203945265689030, 8.770698254607986952513464468418, 9.794794286522626185926083432016, 10.47876343504928953716636541542