| L(s) = 1 | − 2·5-s + 11-s + 2·13-s + 6·17-s − 4·19-s − 6·23-s − 25-s − 6·29-s − 2·31-s − 2·37-s + 10·41-s − 6·43-s + 2·53-s − 2·55-s − 4·59-s + 14·61-s − 4·65-s + 12·67-s + 2·71-s − 6·73-s + 6·83-s − 12·85-s + 6·89-s + 8·95-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 1.25·23-s − 1/5·25-s − 1.11·29-s − 0.359·31-s − 0.328·37-s + 1.56·41-s − 0.914·43-s + 0.274·53-s − 0.269·55-s − 0.520·59-s + 1.79·61-s − 0.496·65-s + 1.46·67-s + 0.237·71-s − 0.702·73-s + 0.658·83-s − 1.30·85-s + 0.635·89-s + 0.820·95-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683101077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683101077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−12.58705427503651, −12.20645617641470, −11.68906801549965, −11.38800745009353, −10.91219709750571, −10.38716497664289, −9.915268732467286, −9.497560892805414, −8.936532830861448, −8.364086331966532, −8.037065935413532, −7.652103731887430, −7.153293063463778, −6.601445760012429, −5.978348507200869, −5.692807778611345, −5.105687200071195, −4.371074773468164, −3.911671829207789, −3.638565939301006, −3.132415282432599, −2.198375923681191, −1.849129025559241, −0.9971781386115127, −0.3927000335601153,
0.3927000335601153, 0.9971781386115127, 1.849129025559241, 2.198375923681191, 3.132415282432599, 3.638565939301006, 3.911671829207789, 4.371074773468164, 5.105687200071195, 5.692807778611345, 5.978348507200869, 6.601445760012429, 7.153293063463778, 7.652103731887430, 8.037065935413532, 8.364086331966532, 8.936532830861448, 9.497560892805414, 9.915268732467286, 10.38716497664289, 10.91219709750571, 11.38800745009353, 11.68906801549965, 12.20645617641470, 12.58705427503651
