| L(s) = 1 | − 7-s − 5·11-s + 3·13-s − 17-s − 6·19-s − 6·23-s + 9·29-s + 4·31-s − 2·37-s + 4·41-s + 10·43-s + 47-s + 49-s + 4·53-s − 8·59-s − 8·61-s + 12·67-s + 8·71-s − 2·73-s + 5·77-s − 13·79-s + 4·83-s − 4·89-s − 3·91-s + 13·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.50·11-s + 0.832·13-s − 0.242·17-s − 1.37·19-s − 1.25·23-s + 1.67·29-s + 0.718·31-s − 0.328·37-s + 0.624·41-s + 1.52·43-s + 0.145·47-s + 1/7·49-s + 0.549·53-s − 1.04·59-s − 1.02·61-s + 1.46·67-s + 0.949·71-s − 0.234·73-s + 0.569·77-s − 1.46·79-s + 0.439·83-s − 0.423·89-s − 0.314·91-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−15.61187162115122, −15.46513164044447, −14.38917743094346, −14.02153076440917, −13.47418841846679, −12.90782262303802, −12.49863684810181, −11.98345504828982, −11.11372678941342, −10.71167924036601, −10.23312669967738, −9.792673910173722, −8.818088260481296, −8.519629443501857, −7.894066134323285, −7.385123841805355, −6.384952882634597, −6.212360122180329, −5.493526687529579, −4.637160004263754, −4.215755831590648, −3.367313963858577, −2.577021740042159, −2.136507482071734, −0.9289715244765153, 0,
0.9289715244765153, 2.136507482071734, 2.577021740042159, 3.367313963858577, 4.215755831590648, 4.637160004263754, 5.493526687529579, 6.212360122180329, 6.384952882634597, 7.385123841805355, 7.894066134323285, 8.519629443501857, 8.818088260481296, 9.792673910173722, 10.23312669967738, 10.71167924036601, 11.11372678941342, 11.98345504828982, 12.49863684810181, 12.90782262303802, 13.47418841846679, 14.02153076440917, 14.38917743094346, 15.46513164044447, 15.61187162115122
