| L(s) = 1 | + 2·5-s − 4·7-s − 3·9-s + 2·13-s + 17-s − 4·19-s − 4·23-s − 25-s − 6·29-s − 4·31-s − 8·35-s + 2·37-s − 6·41-s + 4·43-s − 6·45-s + 9·49-s − 6·53-s − 12·59-s + 10·61-s + 12·63-s + 4·65-s + 4·67-s + 4·71-s − 6·73-s − 12·79-s + 9·81-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 9-s + 0.554·13-s + 0.242·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s − 0.718·31-s − 1.35·35-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 0.894·45-s + 9/7·49-s − 0.824·53-s − 1.56·59-s + 1.28·61-s + 1.51·63-s + 0.496·65-s + 0.488·67-s + 0.474·71-s − 0.702·73-s − 1.35·79-s + 81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−9.446849696394434590254762832260, −8.868215889831003694053677970209, −7.86052886694471390828228448204, −6.64458936246770473314630815851, −6.04868829414574942081046284670, −5.50246131433375739774113763395, −3.94486422776053801827158365621, −3.06428895368232173582353504603, −1.98635588682742520402514989313, 0,
1.98635588682742520402514989313, 3.06428895368232173582353504603, 3.94486422776053801827158365621, 5.50246131433375739774113763395, 6.04868829414574942081046284670, 6.64458936246770473314630815851, 7.86052886694471390828228448204, 8.868215889831003694053677970209, 9.446849696394434590254762832260
