| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 3·11-s − 13-s + 15-s − 17-s − 19-s − 21-s + 6·23-s + 25-s − 27-s + 9·29-s + 2·31-s − 3·33-s − 35-s − 5·37-s + 39-s − 11·41-s + 2·43-s − 45-s − 3·47-s − 6·49-s + 51-s + 9·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s − 0.229·19-s − 0.218·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s + 0.359·31-s − 0.522·33-s − 0.169·35-s − 0.821·37-s + 0.160·39-s − 1.71·41-s + 0.304·43-s − 0.149·45-s − 0.437·47-s − 6/7·49-s + 0.140·51-s + 1.23·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.22570505166489, −12.64191081706289, −12.13644881706014, −11.87755012291822, −11.48044427319253, −10.89563993049190, −10.58701228659000, −9.980398174856395, −9.593288475853711, −8.799438214252887, −8.630338830152033, −8.086051430025349, −7.431893467047266, −6.839226268833845, −6.680039351386167, −6.142845411413619, −5.276418539145346, −5.043415166636158, −4.452068537069173, −4.042321346611951, −3.298653168829178, −2.861142607572490, −2.007281377742635, −1.362218377976528, −0.8304448610035059, 0,
0.8304448610035059, 1.362218377976528, 2.007281377742635, 2.861142607572490, 3.298653168829178, 4.042321346611951, 4.452068537069173, 5.043415166636158, 5.276418539145346, 6.142845411413619, 6.680039351386167, 6.839226268833845, 7.431893467047266, 8.086051430025349, 8.630338830152033, 8.799438214252887, 9.593288475853711, 9.980398174856395, 10.58701228659000, 10.89563993049190, 11.48044427319253, 11.87755012291822, 12.13644881706014, 12.64191081706289, 13.22570505166489
