L(s) = 1 | − 1.48i·2-s + (0.995 − 1.41i)3-s − 0.203·4-s + (0.154 + 0.267i)5-s + (−2.10 − 1.47i)6-s − 2.66i·8-s + (−1.01 − 2.82i)9-s + (0.396 − 0.228i)10-s + (2.73 + 1.58i)11-s + (−0.202 + 0.288i)12-s + (−3.00 − 1.73i)13-s + (0.532 + 0.0472i)15-s − 4.36·16-s + (2.44 + 4.22i)17-s + (−4.18 + 1.51i)18-s + (−4.62 − 2.67i)19-s + ⋯ |
L(s) = 1 | − 1.04i·2-s + (0.574 − 0.818i)3-s − 0.101·4-s + (0.0689 + 0.119i)5-s + (−0.859 − 0.603i)6-s − 0.942i·8-s + (−0.339 − 0.940i)9-s + (0.125 − 0.0723i)10-s + (0.825 + 0.476i)11-s + (−0.0585 + 0.0833i)12-s + (−0.833 − 0.481i)13-s + (0.137 + 0.0121i)15-s − 1.09·16-s + (0.592 + 1.02i)17-s + (−0.987 + 0.356i)18-s + (−1.06 − 0.612i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.615344 - 1.71373i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.615344 - 1.71373i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.995 + 1.41i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.48iT - 2T^{2} \) |
| 5 | \( 1 + (-0.154 - 0.267i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.73 - 1.58i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.00 + 1.73i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.44 - 4.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.62 + 2.67i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.17 + 2.98i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.70 + 1.56i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 7.52iT - 31T^{2} \) |
| 37 | \( 1 + (5.92 - 10.2i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.58 - 4.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.75 - 4.76i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8.46T + 47T^{2} \) |
| 53 | \( 1 + (-0.0740 + 0.0427i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 2.08T + 59T^{2} \) |
| 61 | \( 1 + 5.42iT - 61T^{2} \) |
| 67 | \( 1 + 0.110T + 67T^{2} \) |
| 71 | \( 1 - 7.78iT - 71T^{2} \) |
| 73 | \( 1 + (-8.32 + 4.80i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 5.13T + 79T^{2} \) |
| 83 | \( 1 + (-4.42 - 7.66i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.936 + 1.62i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.9 + 6.34i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.74321103990222042372269488460, −10.07410535720450926202198851096, −9.043694630506872703025161998600, −8.169729722120761462521831526764, −6.88043146481065205250118896258, −6.47820980451605270028134896923, −4.60440007369464588118513131365, −3.29435263514382448121178623155, −2.40351204247986817930669872260, −1.18751243821663121730364149254,
2.30091411769909713622831730025, 3.70865753668393267614157686520, 4.94518589950074050950532841792, 5.73464713965195876119787950968, 7.01523734183642160016343362408, 7.67408481321223575181662816648, 8.921857425092448043956969839485, 9.205488127792750649773304945695, 10.50937790393824992906011432286, 11.32982652847812880745019888082