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
−11.32982652847812880745019888082, −10.50937790393824992906011432286, −9.205488127792750649773304945695, −8.921857425092448043956969839485, −7.67408481321223575181662816648, −7.01523734183642160016343362408, −5.73464713965195876119787950968, −4.94518589950074050950532841792, −3.70865753668393267614157686520, −2.30091411769909713622831730025,
1.18751243821663121730364149254, 2.40351204247986817930669872260, 3.29435263514382448121178623155, 4.60440007369464588118513131365, 6.47820980451605270028134896923, 6.88043146481065205250118896258, 8.169729722120761462521831526764, 9.043694630506872703025161998600, 10.07410535720450926202198851096, 10.74321103990222042372269488460