| L(s) = 1 | + 3-s − 2·5-s + 3·7-s − 2·9-s − 3·11-s − 6·13-s − 2·15-s + 5·17-s − 2·19-s + 3·21-s + 4·23-s − 25-s − 5·27-s − 7·29-s − 5·31-s − 3·33-s − 6·35-s − 11·37-s − 6·39-s − 2·41-s + 8·43-s + 4·45-s + 2·49-s + 5·51-s + 6·53-s + 6·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.13·7-s − 2/3·9-s − 0.904·11-s − 1.66·13-s − 0.516·15-s + 1.21·17-s − 0.458·19-s + 0.654·21-s + 0.834·23-s − 1/5·25-s − 0.962·27-s − 1.29·29-s − 0.898·31-s − 0.522·33-s − 1.01·35-s − 1.80·37-s − 0.960·39-s − 0.312·41-s + 1.21·43-s + 0.596·45-s + 2/7·49-s + 0.700·51-s + 0.824·53-s + 0.809·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 83 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.035733864414314120948556674834, −8.300238805522202750414136632726, −7.54223815052877383551694863706, −7.35216430027834833496476682482, −5.51503883911017029909429289827, −5.09072980037322059662497636841, −3.94512036989515815394609613994, −2.95622302979117959479751854336, −1.96640082330243174820491315964, 0,
1.96640082330243174820491315964, 2.95622302979117959479751854336, 3.94512036989515815394609613994, 5.09072980037322059662497636841, 5.51503883911017029909429289827, 7.35216430027834833496476682482, 7.54223815052877383551694863706, 8.300238805522202750414136632726, 9.035733864414314120948556674834
