| L(s) = 1 | − 5-s − 7-s − 3·9-s + 3·11-s − 4·13-s − 3·17-s + 19-s − 8·23-s − 4·25-s + 2·31-s + 35-s − 8·37-s + 11·43-s + 3·45-s − 7·47-s − 6·49-s + 2·53-s − 3·55-s + 6·59-s − 61-s + 3·63-s + 4·65-s − 10·67-s + 2·71-s + 5·73-s − 3·77-s − 2·79-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s + 0.904·11-s − 1.10·13-s − 0.727·17-s + 0.229·19-s − 1.66·23-s − 4/5·25-s + 0.359·31-s + 0.169·35-s − 1.31·37-s + 1.67·43-s + 0.447·45-s − 1.02·47-s − 6/7·49-s + 0.274·53-s − 0.404·55-s + 0.781·59-s − 0.128·61-s + 0.377·63-s + 0.496·65-s − 1.22·67-s + 0.237·71-s + 0.585·73-s − 0.341·77-s − 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.11217440262524272793966315283, −9.356073062840077109182609673978, −8.467317306890464867116303654123, −7.59019329531238695962577712799, −6.57705024982005416568601729042, −5.71435981136868716408885186688, −4.47863079866238157165170735095, −3.47456597646201355004490006149, −2.19568966813265176024980758112, 0,
2.19568966813265176024980758112, 3.47456597646201355004490006149, 4.47863079866238157165170735095, 5.71435981136868716408885186688, 6.57705024982005416568601729042, 7.59019329531238695962577712799, 8.467317306890464867116303654123, 9.356073062840077109182609673978, 10.11217440262524272793966315283
