L(s) = 1 | + (1 + 1.73i)5-s + (2 − 3.46i)7-s + (2 − 3.46i)11-s + (1 + 1.73i)13-s + 6·17-s − 4·19-s + (0.500 − 0.866i)25-s + (1 − 1.73i)29-s + (−2 − 3.46i)31-s + 7.99·35-s − 2·37-s + (1 + 1.73i)41-s + (−2 + 3.46i)43-s + (4 − 6.92i)47-s + (−4.49 − 7.79i)49-s + ⋯ |
L(s) = 1 | + (0.447 + 0.774i)5-s + (0.755 − 1.30i)7-s + (0.603 − 1.04i)11-s + (0.277 + 0.480i)13-s + 1.45·17-s − 0.917·19-s + (0.100 − 0.173i)25-s + (0.185 − 0.321i)29-s + (−0.359 − 0.622i)31-s + 1.35·35-s − 0.328·37-s + (0.156 + 0.270i)41-s + (−0.304 + 0.528i)43-s + (0.583 − 1.01i)47-s + (−0.642 − 1.11i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.311889180\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.311889180\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{2} \) |
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
−8.690958416420124952169009766158, −7.982401695450692738164788706165, −7.28033858205157524423811543733, −6.45110810502474897836472890513, −5.91626151081979158564460247746, −4.76014148317692414369899770250, −3.89462501042622099339115572877, −3.20482917427896892360203103850, −1.88762344506609723156520019789, −0.847482433613307923310018022962,
1.32433605939148058526627400312, 2.01330341982078390330216007456, 3.18741507589858592513174529470, 4.39786802216298839411257039435, 5.19282704076616678199099315801, 5.63853493754261295942723768777, 6.56280394921089021835490641908, 7.61276724449804036247498814378, 8.326898050841043757046723034398, 8.984679733178239575731899042615