L(s) = 1 | − 5-s + 4·11-s + 6·13-s − 6·17-s + 4·19-s + 8·23-s + 25-s + 10·29-s − 4·31-s + 6·37-s + 6·41-s − 4·43-s − 12·47-s + 6·53-s − 4·55-s − 4·59-s − 2·61-s − 6·65-s − 4·67-s + 2·73-s − 8·79-s + 12·83-s + 6·85-s + 14·89-s − 4·95-s − 6·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.20·11-s + 1.66·13-s − 1.45·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.85·29-s − 0.718·31-s + 0.986·37-s + 0.937·41-s − 0.609·43-s − 1.75·47-s + 0.824·53-s − 0.539·55-s − 0.520·59-s − 0.256·61-s − 0.744·65-s − 0.488·67-s + 0.234·73-s − 0.900·79-s + 1.31·83-s + 0.650·85-s + 1.48·89-s − 0.410·95-s − 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.531953728\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.531953728\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.47680576920250, −12.98124182705466, −12.49033600114678, −11.78509159349358, −11.43524845871763, −11.15774073791142, −10.67200558008967, −10.08006983868577, −9.311047273796885, −8.968040896089132, −8.736875622997918, −8.071270092412278, −7.560810597222119, −6.786124485056770, −6.569867894140331, −6.167860762362742, −5.387872388661633, −4.709967395528193, −4.359434954288177, −3.702364311368292, −3.233756487637736, −2.688277248944856, −1.720874698854997, −1.137990593205025, −0.6505288729915996,
0.6505288729915996, 1.137990593205025, 1.720874698854997, 2.688277248944856, 3.233756487637736, 3.702364311368292, 4.359434954288177, 4.709967395528193, 5.387872388661633, 6.167860762362742, 6.569867894140331, 6.786124485056770, 7.560810597222119, 8.071270092412278, 8.736875622997918, 8.968040896089132, 9.311047273796885, 10.08006983868577, 10.67200558008967, 11.15774073791142, 11.43524845871763, 11.78509159349358, 12.49033600114678, 12.98124182705466, 13.47680576920250