L(s) = 1 | + (−0.951 + 1.04i)2-s + (−1.66 − 0.475i)3-s + (−0.190 − 1.99i)4-s + (−2.36 − 1.36i)5-s + (2.08 − 1.29i)6-s + (−1.89 − 1.84i)7-s + (2.26 + 1.69i)8-s + (2.54 + 1.58i)9-s + (3.67 − 1.17i)10-s + (−1.02 − 1.76i)11-s + (−0.629 + 3.40i)12-s − 4.44·13-s + (3.73 − 0.232i)14-s + (3.28 + 3.39i)15-s + (−3.92 + 0.759i)16-s + (0.571 − 0.330i)17-s + ⋯ |
L(s) = 1 | + (−0.672 + 0.740i)2-s + (−0.961 − 0.274i)3-s + (−0.0953 − 0.995i)4-s + (−1.05 − 0.609i)5-s + (0.850 − 0.526i)6-s + (−0.717 − 0.696i)7-s + (0.800 + 0.598i)8-s + (0.849 + 0.528i)9-s + (1.16 − 0.371i)10-s + (−0.307 − 0.532i)11-s + (−0.181 + 0.983i)12-s − 1.23·13-s + (0.998 − 0.0621i)14-s + (0.848 + 0.876i)15-s + (−0.981 + 0.189i)16-s + (0.138 − 0.0800i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.129302 - 0.188199i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.129302 - 0.188199i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 - 1.04i)T \) |
| 3 | \( 1 + (1.66 + 0.475i)T \) |
| 7 | \( 1 + (1.89 + 1.84i)T \) |
good | 5 | \( 1 + (2.36 + 1.36i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.02 + 1.76i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 + (-0.571 + 0.330i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.54 - 1.46i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.521 - 0.902i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.06iT - 29T^{2} \) |
| 31 | \( 1 + (-2.68 + 1.55i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.76 + 8.24i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 7.85iT - 41T^{2} \) |
| 43 | \( 1 - 0.530iT - 43T^{2} \) |
| 47 | \( 1 + (-0.521 + 0.902i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (9.16 - 5.29i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.74 + 3.02i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.0992 + 0.171i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.12 - 2.38i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 + (-1.51 - 2.62i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.45 + 5.45i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.15T + 83T^{2} \) |
| 89 | \( 1 + (-0.468 - 0.270i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6.91T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.01797074911488834338706685719, −12.76711563904358168024687187285, −11.72677075544670355869658786186, −10.57137686183606866139053829192, −9.527133299110465595441481523914, −7.87323897659061769395350170769, −7.21630074724585026742839342327, −5.79027101294043872916292763649, −4.44836422611681104490021284289, −0.38989777422332225384693281936,
3.06372709978767137127841090385, 4.70673043403025851114978044802, 6.73330909686000633933326663238, 7.77521973762259366965971498638, 9.486901898008324671721528379356, 10.26619652462212347995236309093, 11.48600519683683376007117569724, 12.06379839119806584023692328078, 12.92391662817802070954498010258, 14.98823111221846710588514583196