L(s) = 1 | − 5-s + 4·7-s + 4·17-s − 8·19-s + 23-s + 25-s − 2·29-s − 4·31-s − 4·35-s − 2·37-s + 8·41-s + 4·43-s − 8·47-s + 9·49-s + 6·53-s − 10·59-s − 8·61-s − 12·67-s − 16·71-s − 2·73-s + 6·79-s + 2·83-s − 4·85-s + 6·89-s + 8·95-s − 10·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s + 0.970·17-s − 1.83·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.676·35-s − 0.328·37-s + 1.24·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.824·53-s − 1.30·59-s − 1.02·61-s − 1.46·67-s − 1.89·71-s − 0.234·73-s + 0.675·79-s + 0.219·83-s − 0.433·85-s + 0.635·89-s + 0.820·95-s − 1.01·97-s + 0.0995·101-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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 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
−16.42485098787619, −15.46117603938311, −14.91519808518678, −14.68719561372697, −14.18597800700761, −13.42427530163235, −12.81310972660043, −12.20584843111888, −11.78907067425055, −10.95900647044824, −10.83688509277812, −10.17073923049421, −9.182334392956256, −8.783108930334446, −8.078533494802232, −7.681423253245888, −7.148837699269189, −6.182389700568913, −5.665687699092563, −4.805275129618350, −4.413133067615434, −3.708999203632908, −2.779721912011486, −1.897145347309351, −1.260579108173776, 0,
1.260579108173776, 1.897145347309351, 2.779721912011486, 3.708999203632908, 4.413133067615434, 4.805275129618350, 5.665687699092563, 6.182389700568913, 7.148837699269189, 7.681423253245888, 8.078533494802232, 8.783108930334446, 9.182334392956256, 10.17073923049421, 10.83688509277812, 10.95900647044824, 11.78907067425055, 12.20584843111888, 12.81310972660043, 13.42427530163235, 14.18597800700761, 14.68719561372697, 14.91519808518678, 15.46117603938311, 16.42485098787619