| L(s) = 1 | + 5-s + 2·11-s − 6·17-s − 4·19-s + 23-s + 25-s + 8·31-s − 6·37-s + 2·41-s + 2·43-s + 4·47-s − 7·49-s + 2·53-s + 2·55-s − 2·61-s + 2·67-s − 10·71-s − 10·73-s − 4·83-s − 6·85-s − 4·89-s − 4·95-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.603·11-s − 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.43·31-s − 0.986·37-s + 0.312·41-s + 0.304·43-s + 0.583·47-s − 49-s + 0.274·53-s + 0.269·55-s − 0.256·61-s + 0.244·67-s − 1.18·71-s − 1.17·73-s − 0.439·83-s − 0.650·85-s − 0.423·89-s − 0.410·95-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−16.09322070412962, −15.61092234323097, −15.17304379306690, −14.39909573677424, −14.10762500537293, −13.25400189967981, −13.09627961441375, −12.29856495465432, −11.70163563890356, −11.18712390026014, −10.48667429818355, −10.11756679301990, −9.261974685852125, −8.828305005549434, −8.389355144524387, −7.487217899165067, −6.807433150519174, −6.340528174912121, −5.812586296650457, −4.786774850261952, −4.443064266323669, −3.618759255526068, −2.693599393000127, −2.065257532561675, −1.200245805734309, 0,
1.200245805734309, 2.065257532561675, 2.693599393000127, 3.618759255526068, 4.443064266323669, 4.786774850261952, 5.812586296650457, 6.340528174912121, 6.807433150519174, 7.487217899165067, 8.389355144524387, 8.828305005549434, 9.261974685852125, 10.11756679301990, 10.48667429818355, 11.18712390026014, 11.70163563890356, 12.29856495465432, 13.09627961441375, 13.25400189967981, 14.10762500537293, 14.39909573677424, 15.17304379306690, 15.61092234323097, 16.09322070412962
