L(s) = 1 | − 2·5-s − 2·11-s + 4·13-s − 17-s + 2·19-s − 6·23-s − 25-s + 4·31-s + 2·37-s + 6·41-s − 12·43-s − 12·47-s − 10·53-s + 4·55-s + 4·59-s + 2·61-s − 8·65-s − 12·67-s − 2·71-s + 14·73-s + 8·79-s + 4·83-s + 2·85-s + 18·89-s − 4·95-s + 10·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.603·11-s + 1.10·13-s − 0.242·17-s + 0.458·19-s − 1.25·23-s − 1/5·25-s + 0.718·31-s + 0.328·37-s + 0.937·41-s − 1.82·43-s − 1.75·47-s − 1.37·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s − 0.992·65-s − 1.46·67-s − 0.237·71-s + 1.63·73-s + 0.900·79-s + 0.439·83-s + 0.216·85-s + 1.90·89-s − 0.410·95-s + 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−13.62762633819338, −13.35167570213284, −12.96451709196248, −12.21990919378944, −11.79563665139810, −11.52400370011956, −10.89114674156579, −10.55966851199697, −9.805938752471475, −9.580841447122850, −8.733709008475798, −8.300420557367274, −7.896511236463932, −7.597431384411136, −6.815082617191856, −6.125009743848193, −6.068788061107420, −4.911282050458335, −4.851899656835780, −3.938697278450580, −3.557359975199127, −3.077442279798105, −2.210519536661692, −1.610283256200408, −0.7343794416947569, 0,
0.7343794416947569, 1.610283256200408, 2.210519536661692, 3.077442279798105, 3.557359975199127, 3.938697278450580, 4.851899656835780, 4.911282050458335, 6.068788061107420, 6.125009743848193, 6.815082617191856, 7.597431384411136, 7.896511236463932, 8.300420557367274, 8.733709008475798, 9.580841447122850, 9.805938752471475, 10.55966851199697, 10.89114674156579, 11.52400370011956, 11.79563665139810, 12.21990919378944, 12.96451709196248, 13.35167570213284, 13.62762633819338