L(s) = 1 | + (0.965 − 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + (1.55 − 1.61i)5-s − i·6-s + (−1.38 + 2.25i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.08 − 1.95i)10-s + (0.582 + 1.00i)11-s + (−0.258 − 0.965i)12-s + (−1.92 − 1.92i)13-s + (−0.756 + 2.53i)14-s + (−1.15 − 1.91i)15-s + (0.500 − 0.866i)16-s + (−0.0209 − 0.00560i)17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + (0.693 − 0.720i)5-s − 0.408i·6-s + (−0.524 + 0.851i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.341 − 0.618i)10-s + (0.175 + 0.304i)11-s + (−0.0747 − 0.278i)12-s + (−0.533 − 0.533i)13-s + (−0.202 + 0.677i)14-s + (−0.298 − 0.494i)15-s + (0.125 − 0.216i)16-s + (−0.00507 − 0.00136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.69054 - 0.807613i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69054 - 0.807613i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 + (-1.55 + 1.61i)T \) |
| 7 | \( 1 + (1.38 - 2.25i)T \) |
good | 11 | \( 1 + (-0.582 - 1.00i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.92 + 1.92i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.0209 + 0.00560i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.989 - 1.71i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.85 - 6.93i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 5.60iT - 29T^{2} \) |
| 31 | \( 1 + (-6.86 + 3.96i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (10.2 - 2.74i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 2.48iT - 41T^{2} \) |
| 43 | \( 1 + (7.87 - 7.87i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.05 + 3.94i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.82 - 0.757i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-5.34 - 9.25i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.15 + 1.82i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.76 + 14.0i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 7.51T + 71T^{2} \) |
| 73 | \( 1 + (-0.969 + 3.61i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (1.39 + 0.805i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (9.74 + 9.74i)T + 83iT^{2} \) |
| 89 | \( 1 + (-1.80 + 3.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.265 - 0.265i)T - 97iT^{2} \) |
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
−12.37318289980475665879007120223, −11.75767495984211451383110584054, −10.20030085211047380719584911519, −9.349548160185982195062605827633, −8.274980624013581445757433302812, −6.88523941148322900271997244728, −5.82299536761653644859280524314, −4.98704420479670155194853037240, −3.16561949050084236796912819948, −1.79485023929811537208191051065,
2.58128882892844727584666324334, 3.80790977953251151354013237807, 5.00498983295720911561475008730, 6.42893678217930556832848627831, 7.01488732137256503563483475314, 8.570792031456780329175306723966, 9.873201904438100315121964087448, 10.48228273542813756989519567157, 11.47393535835881726280186725998, 12.70288707566503090606552962892