L(s) = 1 | + 2·5-s − 7-s − 3·9-s + 4·11-s − 2·13-s + 6·17-s + 8·19-s − 25-s − 6·29-s − 2·35-s + 2·37-s + 2·41-s + 4·43-s − 6·45-s − 8·47-s + 49-s − 6·53-s + 8·55-s + 6·61-s + 3·63-s − 4·65-s − 4·67-s − 8·71-s − 10·73-s − 4·77-s − 16·79-s + 9·81-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s − 9-s + 1.20·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s − 0.338·35-s + 0.328·37-s + 0.312·41-s + 0.609·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s + 0.768·61-s + 0.377·63-s − 0.496·65-s − 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.455·77-s − 1.80·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 31 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.69608394342752, −14.25506615856019, −13.72828394461293, −13.30211162726050, −12.63019127042617, −12.02982234414309, −11.68384021312478, −11.28745057601789, −10.47737409216275, −9.840041962243111, −9.427059902824213, −9.358631617757358, −8.481072963221965, −7.846174628401750, −7.312725742907878, −6.780866079629476, −5.851157965429278, −5.811745111399666, −5.275379869345427, −4.424770626794890, −3.601884931192213, −3.131192288160578, −2.582378759313984, −1.589561932895029, −1.137613368655702, 0,
1.137613368655702, 1.589561932895029, 2.582378759313984, 3.131192288160578, 3.601884931192213, 4.424770626794890, 5.275379869345427, 5.811745111399666, 5.851157965429278, 6.780866079629476, 7.312725742907878, 7.846174628401750, 8.481072963221965, 9.358631617757358, 9.427059902824213, 9.840041962243111, 10.47737409216275, 11.28745057601789, 11.68384021312478, 12.02982234414309, 12.63019127042617, 13.30211162726050, 13.72828394461293, 14.25506615856019, 14.69608394342752