L(s) = 1 | + 2.30i·2-s + (0.736 + 1.27i)3-s − 3.30·4-s + (0.733 − 0.423i)5-s + (−2.93 + 1.69i)6-s + (−0.357 − 2.62i)7-s − 3.00i·8-s + (0.414 − 0.718i)9-s + (0.975 + 1.69i)10-s + (1.30 − 0.751i)11-s + (−2.43 − 4.21i)12-s + (−2.92 + 2.11i)13-s + (6.03 − 0.824i)14-s + (1.08 + 0.624i)15-s + 0.313·16-s − 2.07·17-s + ⋯ |
L(s) = 1 | + 1.62i·2-s + (0.425 + 0.736i)3-s − 1.65·4-s + (0.328 − 0.189i)5-s + (−1.19 + 0.692i)6-s + (−0.135 − 0.990i)7-s − 1.06i·8-s + (0.138 − 0.239i)9-s + (0.308 + 0.534i)10-s + (0.392 − 0.226i)11-s + (−0.702 − 1.21i)12-s + (−0.810 + 0.585i)13-s + (1.61 − 0.220i)14-s + (0.279 + 0.161i)15-s + 0.0782·16-s − 0.502·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.445680 + 0.971294i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.445680 + 0.971294i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.357 + 2.62i)T \) |
| 13 | \( 1 + (2.92 - 2.11i)T \) |
good | 2 | \( 1 - 2.30iT - 2T^{2} \) |
| 3 | \( 1 + (-0.736 - 1.27i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.733 + 0.423i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.30 + 0.751i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 + (-0.0410 - 0.0237i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 7.81T + 23T^{2} \) |
| 29 | \( 1 + (0.679 - 1.17i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (6.80 + 3.93i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.70iT - 37T^{2} \) |
| 41 | \( 1 + (8.67 + 5.00i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.63 - 8.02i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.311 - 0.180i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.35 - 2.34i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 1.64iT - 59T^{2} \) |
| 61 | \( 1 + (2.26 - 3.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.76 - 1.02i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.3 + 7.10i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.85 - 3.38i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.82 + 10.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 11.5iT - 83T^{2} \) |
| 89 | \( 1 - 17.5iT - 89T^{2} \) |
| 97 | \( 1 + (-0.369 + 0.213i)T + (48.5 - 84.0i)T^{2} \) |
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
−14.69259040322991413121029658272, −13.91526529627736121929986990324, −12.90140203380108369188756611121, −11.01250540777947229221813818663, −9.507970314615427977720799930567, −9.028850275180883122244604245973, −7.46868908341874741275798635430, −6.64402339353704994265419226554, −5.08561285753788235415578127806, −3.93187325209653416162588145160,
1.93215821454881907906339766027, 2.98211847019348852409088570769, 4.98267779755160012875900137534, 6.89838642056512864900644532493, 8.511410383031262519802813124368, 9.524270248362346884155124418671, 10.56195323780077872098810316459, 11.74347807521771168101617378413, 12.62639870109337654898105823890, 13.21098375555296210455720344612