L(s) = 1 | − 2·3-s − 2·5-s − 7-s + 9-s − 4·13-s + 4·15-s − 6·19-s + 2·21-s − 3·23-s − 25-s + 4·27-s + 5·29-s + 2·31-s + 2·35-s − 3·37-s + 8·39-s + 7·43-s − 2·45-s + 49-s − 53-s + 12·57-s − 14·59-s + 2·61-s − 63-s + 8·65-s + 4·67-s + 6·69-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s + 1.03·15-s − 1.37·19-s + 0.436·21-s − 0.625·23-s − 1/5·25-s + 0.769·27-s + 0.928·29-s + 0.359·31-s + 0.338·35-s − 0.493·37-s + 1.28·39-s + 1.06·43-s − 0.298·45-s + 1/7·49-s − 0.137·53-s + 1.58·57-s − 1.82·59-s + 0.256·61-s − 0.125·63-s + 0.992·65-s + 0.488·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.22347526775205, −15.70163432932713, −15.36819130737939, −14.58007535396992, −14.12400204419145, −13.35619744084764, −12.45898479210972, −12.32871873807844, −11.90216927763416, −11.14947711349938, −10.75147547723835, −10.16342296507661, −9.558142122095044, −8.752621787450860, −8.105859008672411, −7.567398598222871, −6.786498870070170, −6.352082764406503, −5.728179887366726, −4.902778073077779, −4.462690709338290, −3.761943169204186, −2.836775459615547, −2.035762536657411, −0.6826738945641955, 0,
0.6826738945641955, 2.035762536657411, 2.836775459615547, 3.761943169204186, 4.462690709338290, 4.902778073077779, 5.728179887366726, 6.352082764406503, 6.786498870070170, 7.567398598222871, 8.105859008672411, 8.752621787450860, 9.558142122095044, 10.16342296507661, 10.75147547723835, 11.14947711349938, 11.90216927763416, 12.32871873807844, 12.45898479210972, 13.35619744084764, 14.12400204419145, 14.58007535396992, 15.36819130737939, 15.70163432932713, 16.22347526775205