L(s) = 1 | + (−1.36 + 1.36i)2-s + (1.5 + 0.866i)3-s − 1.73i·4-s + (−0.133 − 2.23i)5-s + (−3.23 + 0.866i)6-s + (2 − 1.73i)7-s + (−0.366 − 0.366i)8-s + (1.5 + 2.59i)9-s + (3.23 + 2.86i)10-s + (2.73 − 4.73i)11-s + (1.49 − 2.59i)12-s + (1 + 3.73i)13-s + (−0.366 + 5.09i)14-s + (1.73 − 3.46i)15-s + 4.46·16-s + (−0.732 + 2.73i)17-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.965i)2-s + (0.866 + 0.499i)3-s − 0.866i·4-s + (−0.0599 − 0.998i)5-s + (−1.31 + 0.353i)6-s + (0.755 − 0.654i)7-s + (−0.129 − 0.129i)8-s + (0.5 + 0.866i)9-s + (1.02 + 0.906i)10-s + (0.823 − 1.42i)11-s + (0.433 − 0.749i)12-s + (0.277 + 1.03i)13-s + (−0.0978 + 1.36i)14-s + (0.447 − 0.894i)15-s + 1.11·16-s + (−0.177 + 0.662i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06365 + 0.444375i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06365 + 0.444375i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 + (0.133 + 2.23i)T \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 2 | \( 1 + (1.36 - 1.36i)T - 2iT^{2} \) |
| 11 | \( 1 + (-2.73 + 4.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 - 3.73i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (0.732 - 2.73i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.63 + 2.83i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.23 + 4.59i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (6 - 3.46i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 0.196iT - 31T^{2} \) |
| 37 | \( 1 + (-1.09 - 4.09i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.19 + 1.26i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.401 - 1.5i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-9.29 - 9.29i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.63 - 6.09i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 11.3iT - 61T^{2} \) |
| 67 | \( 1 + (-0.901 + 0.901i)T - 67iT^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (8.83 + 2.36i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 - 4.53iT - 79T^{2} \) |
| 83 | \( 1 + (1.36 + 0.366i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-4.59 + 7.96i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.09 + 7.83i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.50774686754646020295724218996, −10.59820031666879158039119379015, −9.250991118974983309380382146040, −8.895630326548943972802933167358, −8.236482186205813697222391911248, −7.33384513408846383350110244655, −6.11969512133881678930208071563, −4.66775759763220961767394829433, −3.65635652176926386080733605700, −1.29028602696991915967275935970,
1.65257937697496838385957618899, 2.54387019424675151150326865726, 3.73052820374428214637474593329, 5.71244270739327673230280687041, 7.24215249684585579332716434953, 7.83013209222033954948963273177, 8.955645839324955095374697287038, 9.652733745412925308586662510163, 10.44634354054030879698209636522, 11.65156565204422681705779517961