L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (0.500 + 0.866i)4-s + 0.999·6-s + (−0.5 + 2.59i)7-s − 3·8-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)12-s − 3·13-s + (−2 − 1.73i)14-s + (0.500 − 0.866i)16-s + (−1 − 1.73i)17-s + (−0.499 − 0.866i)18-s + (−0.5 + 0.866i)19-s + (2.5 − 0.866i)21-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (0.250 + 0.433i)4-s + 0.408·6-s + (−0.188 + 0.981i)7-s − 1.06·8-s + (−0.166 + 0.288i)9-s + (0.144 − 0.249i)12-s − 0.832·13-s + (−0.534 − 0.462i)14-s + (0.125 − 0.216i)16-s + (−0.242 − 0.420i)17-s + (−0.117 − 0.204i)18-s + (−0.114 + 0.198i)19-s + (0.545 − 0.188i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0361939 + 0.570341i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0361939 + 0.570341i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7 - 12.1i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−11.67452797773668899795018965322, −10.31024721841237927407162233931, −9.212393169500237287876323174469, −8.539417939631062783659611063785, −7.57908592514341036045344572855, −6.85725210751749232366599614970, −5.97005877461043110162735847241, −5.04214255537010575754896869277, −3.28044334535376912751289032626, −2.16268097790819128946829785936,
0.35913349243608183229805716791, 2.08118567249203508341797639570, 3.49746741578775272329453905956, 4.60605246366628147672950064710, 5.78281058328925448121374256436, 6.71344280614133355395666442400, 7.73737338456696699061634793189, 9.080069573372687282623337245520, 9.779564189415215858255571069228, 10.46239083820504489574384219663