| L(s) = 1 | + 3-s + 2·5-s + 7-s − 2·9-s + 2·11-s − 2·13-s + 2·15-s + 5·19-s + 21-s − 4·23-s − 25-s − 5·27-s − 29-s − 31-s + 2·33-s + 2·35-s + 10·37-s − 2·39-s − 8·41-s − 4·43-s − 4·45-s − 3·47-s + 49-s + 3·53-s + 4·55-s + 5·57-s + 11·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s − 2/3·9-s + 0.603·11-s − 0.554·13-s + 0.516·15-s + 1.14·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.962·27-s − 0.185·29-s − 0.179·31-s + 0.348·33-s + 0.338·35-s + 1.64·37-s − 0.320·39-s − 1.24·41-s − 0.609·43-s − 0.596·45-s − 0.437·47-s + 1/7·49-s + 0.412·53-s + 0.539·55-s + 0.662·57-s + 1.43·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.71729123790259, −13.49942702251586, −12.91430409385294, −12.18011775388798, −11.72710923666405, −11.51844451714121, −10.83587848881616, −10.12602227562203, −9.800610947217805, −9.395832351807764, −8.909683818717066, −8.413043141480992, −7.748377044315698, −7.567748628811291, −6.712361349425058, −6.253445330153514, −5.652364734027687, −5.334058400253237, −4.645504458215964, −4.010683796473942, −3.353676920436616, −2.862695533418586, −2.154683035844805, −1.777345470182077, −1.003487528360934, 0,
1.003487528360934, 1.777345470182077, 2.154683035844805, 2.862695533418586, 3.353676920436616, 4.010683796473942, 4.645504458215964, 5.334058400253237, 5.652364734027687, 6.253445330153514, 6.712361349425058, 7.567748628811291, 7.748377044315698, 8.413043141480992, 8.909683818717066, 9.395832351807764, 9.800610947217805, 10.12602227562203, 10.83587848881616, 11.51844451714121, 11.72710923666405, 12.18011775388798, 12.91430409385294, 13.49942702251586, 13.71729123790259
