L(s) = 1 | − 5-s − 4·11-s + 13-s + 4·17-s − 2·19-s + 4·23-s + 25-s − 6·29-s + 2·31-s − 8·37-s + 2·41-s + 2·43-s + 10·47-s + 6·53-s + 4·55-s − 12·59-s − 8·61-s − 65-s + 12·67-s + 8·71-s + 6·73-s − 8·79-s − 14·83-s − 4·85-s + 6·89-s + 2·95-s + 10·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.20·11-s + 0.277·13-s + 0.970·17-s − 0.458·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s − 1.31·37-s + 0.312·41-s + 0.304·43-s + 1.45·47-s + 0.824·53-s + 0.539·55-s − 1.56·59-s − 1.02·61-s − 0.124·65-s + 1.46·67-s + 0.949·71-s + 0.702·73-s − 0.900·79-s − 1.53·83-s − 0.433·85-s + 0.635·89-s + 0.205·95-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.06545957088416, −12.73857166628725, −12.20628379228974, −11.92546114482343, −11.14890961771181, −10.81383241303184, −10.54143525523198, −9.959015182701485, −9.378931573167453, −8.965191298966753, −8.334630485233727, −8.045708076219244, −7.366045310683063, −7.243607355952132, −6.502138981095004, −5.867854586326899, −5.405514344476114, −5.066084361340292, −4.359451863736070, −3.838866243165070, −3.283724760201430, −2.774138895726801, −2.188316411549513, −1.440705125077949, −0.7248821164345109, 0,
0.7248821164345109, 1.440705125077949, 2.188316411549513, 2.774138895726801, 3.283724760201430, 3.838866243165070, 4.359451863736070, 5.066084361340292, 5.405514344476114, 5.867854586326899, 6.502138981095004, 7.243607355952132, 7.366045310683063, 8.045708076219244, 8.334630485233727, 8.965191298966753, 9.378931573167453, 9.959015182701485, 10.54143525523198, 10.81383241303184, 11.14890961771181, 11.92546114482343, 12.20628379228974, 12.73857166628725, 13.06545957088416