| L(s) = 1 | + 5-s − 7-s − 2·11-s + 13-s − 7·19-s + 3·23-s − 4·25-s + 9·29-s + 5·31-s − 35-s − 8·37-s + 10·41-s + 5·43-s − 7·47-s + 49-s − 3·53-s − 2·55-s + 6·61-s + 65-s − 10·67-s − 4·71-s − 11·73-s + 2·77-s − 11·79-s − 11·83-s + 3·89-s − 91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.603·11-s + 0.277·13-s − 1.60·19-s + 0.625·23-s − 4/5·25-s + 1.67·29-s + 0.898·31-s − 0.169·35-s − 1.31·37-s + 1.56·41-s + 0.762·43-s − 1.02·47-s + 1/7·49-s − 0.412·53-s − 0.269·55-s + 0.768·61-s + 0.124·65-s − 1.22·67-s − 0.474·71-s − 1.28·73-s + 0.227·77-s − 1.23·79-s − 1.20·83-s + 0.317·89-s − 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−7.66914847156675308748810502522, −6.83772489665389638417589680887, −6.23191861913624368915142545331, −5.66888164355744385523347126110, −4.70462681641584582462195219492, −4.12556278942285630433767071210, −3.00445400946739823266676932889, −2.41688276387460613412501930136, −1.33304308048761542490292123682, 0,
1.33304308048761542490292123682, 2.41688276387460613412501930136, 3.00445400946739823266676932889, 4.12556278942285630433767071210, 4.70462681641584582462195219492, 5.66888164355744385523347126110, 6.23191861913624368915142545331, 6.83772489665389638417589680887, 7.66914847156675308748810502522
