L(s) = 1 | + (−1.60 + 0.656i)3-s − 0.670i·5-s + (2.13 − 2.10i)9-s − 5.24·11-s + 2·13-s + (0.440 + 1.07i)15-s + 4.21i·17-s + 1.31i·19-s + 3.20·23-s + 4.54·25-s + (−2.04 + 4.77i)27-s − 6.22i·29-s − 1.73i·31-s + (8.41 − 3.44i)33-s − 6.27·37-s + ⋯ |
L(s) = 1 | + (−0.925 + 0.379i)3-s − 0.300i·5-s + (0.712 − 0.701i)9-s − 1.58·11-s + 0.554·13-s + (0.113 + 0.277i)15-s + 1.02i·17-s + 0.301i·19-s + 0.668·23-s + 0.909·25-s + (−0.393 + 0.919i)27-s − 1.15i·29-s − 0.311i·31-s + (1.46 − 0.600i)33-s − 1.03·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6153299973\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6153299973\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.60 - 0.656i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.670iT - 5T^{2} \) |
| 11 | \( 1 + 5.24T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 4.21iT - 17T^{2} \) |
| 19 | \( 1 - 1.31iT - 19T^{2} \) |
| 23 | \( 1 - 3.20T + 23T^{2} \) |
| 29 | \( 1 + 6.22iT - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 6.27T + 37T^{2} \) |
| 41 | \( 1 - 9.76iT - 41T^{2} \) |
| 43 | \( 1 + 9.55iT - 43T^{2} \) |
| 47 | \( 1 + 9.61T + 47T^{2} \) |
| 53 | \( 1 + 11.7iT - 53T^{2} \) |
| 59 | \( 1 - 7.57T + 59T^{2} \) |
| 61 | \( 1 + 3.72T + 61T^{2} \) |
| 67 | \( 1 - 2.98iT - 67T^{2} \) |
| 71 | \( 1 + 4.08T + 71T^{2} \) |
| 73 | \( 1 - 0.274T + 73T^{2} \) |
| 79 | \( 1 - 5.19iT - 79T^{2} \) |
| 83 | \( 1 + 2.04T + 83T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 + 3.27T + 97T^{2} \) |
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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
−8.648766783511081719142385389131, −8.140675530817874648984370019935, −7.12567817586322386324060951638, −6.30841622481549009465596278522, −5.51850737393959745610542074912, −4.95933461295882409248505784854, −4.05663824237775057188067815145, −3.05926442212231722707375156643, −1.66139833647787902068473503064, −0.27205766481179060776121652906,
1.08638968794386640613120141256, 2.45109175512055151720688752235, 3.32333587827040788653209079650, 4.86413838988026230622093629844, 5.10933435281621584662505662512, 6.07665000405897029177346587242, 6.98963834725337290915516112582, 7.38842768166198492172873146860, 8.329180763925792531481069408397, 9.177880595262805660021459494206