| L(s) = 1 | + (−0.208 − 0.00838i)2-s + (−0.278 + 0.960i)3-s + (−1.95 − 0.157i)4-s + (−1.53 − 0.583i)5-s + (0.0659 − 0.197i)6-s + (−0.815 − 1.91i)7-s + (0.818 + 0.0993i)8-s + (−0.845 − 0.534i)9-s + (0.315 + 0.134i)10-s + (1.34 + 2.13i)11-s + (0.693 − 1.82i)12-s + (1.58 + 3.23i)13-s + (0.153 + 0.405i)14-s + (0.987 − 1.31i)15-s + (3.69 + 0.600i)16-s + (5.49 − 2.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.147 − 0.00593i)2-s + (−0.160 + 0.554i)3-s + (−0.975 − 0.0787i)4-s + (−0.687 − 0.260i)5-s + (0.0269 − 0.0806i)6-s + (−0.308 − 0.723i)7-s + (0.289 + 0.0351i)8-s + (−0.281 − 0.178i)9-s + (0.0996 + 0.0424i)10-s + (0.406 + 0.643i)11-s + (0.200 − 0.528i)12-s + (0.439 + 0.898i)13-s + (0.0410 + 0.108i)14-s + (0.255 − 0.339i)15-s + (0.923 + 0.150i)16-s + (1.33 − 0.567i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.864140 + 0.0693763i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.864140 + 0.0693763i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.278 - 0.960i)T \) |
| 13 | \( 1 + (-1.58 - 3.23i)T \) |
| good | 2 | \( 1 + (0.208 + 0.00838i)T + (1.99 + 0.160i)T^{2} \) |
| 5 | \( 1 + (1.53 + 0.583i)T + (3.74 + 3.31i)T^{2} \) |
| 7 | \( 1 + (0.815 + 1.91i)T + (-4.84 + 5.04i)T^{2} \) |
| 11 | \( 1 + (-1.34 - 2.13i)T + (-4.71 + 9.93i)T^{2} \) |
| 17 | \( 1 + (-5.49 + 2.34i)T + (11.7 - 12.2i)T^{2} \) |
| 19 | \( 1 + (0.388 - 0.224i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.86 + 6.69i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.198 - 4.92i)T + (-28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (-4.74 - 5.36i)T + (-3.73 + 30.7i)T^{2} \) |
| 37 | \( 1 + (-8.46 + 1.72i)T + (34.0 - 14.5i)T^{2} \) |
| 41 | \( 1 + (-2.43 - 0.705i)T + (34.6 + 21.9i)T^{2} \) |
| 43 | \( 1 + (1.69 - 8.28i)T + (-39.5 - 16.8i)T^{2} \) |
| 47 | \( 1 + (-1.89 + 1.30i)T + (16.6 - 43.9i)T^{2} \) |
| 53 | \( 1 + (-0.771 + 6.35i)T + (-51.4 - 12.6i)T^{2} \) |
| 59 | \( 1 + (1.36 + 8.40i)T + (-55.9 + 18.6i)T^{2} \) |
| 61 | \( 1 + (10.2 - 7.72i)T + (16.9 - 58.5i)T^{2} \) |
| 67 | \( 1 + (-0.919 - 11.3i)T + (-66.1 + 10.7i)T^{2} \) |
| 71 | \( 1 + (-10.8 + 10.3i)T + (2.85 - 70.9i)T^{2} \) |
| 73 | \( 1 + (-0.199 + 0.379i)T + (-41.4 - 60.0i)T^{2} \) |
| 79 | \( 1 + (8.80 + 12.7i)T + (-28.0 + 73.8i)T^{2} \) |
| 83 | \( 1 + (-2.34 + 9.53i)T + (-73.4 - 38.5i)T^{2} \) |
| 89 | \( 1 + (-14.7 - 8.51i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.01 - 4.09i)T + (19.4 + 95.0i)T^{2} \) |
| 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
−10.73502951407730274029333078173, −9.965140286743696670540943051778, −9.246106812868033584496267733205, −8.412595247668007326489126252654, −7.43338906641410936187286964252, −6.33050451589026710468552127953, −4.85445710718832893634147153367, −4.34148892689959392114132893960, −3.37803939150795790981648989535, −0.898337788543623820227341573121,
0.922988082584719200622009656292, 3.06785715775811824236526062432, 3.95123378676884247383966814183, 5.50542816373119680720528399645, 6.04397787270281932051302461703, 7.65453809000826013561364589920, 8.026438281455899081841112708812, 9.050780310547517724543891075371, 9.859925991178386137304162217691, 11.00353670828385778375702144087
