| L(s) = 1 | − 9-s − 12·11-s − 2·19-s + 12·29-s − 6·31-s + 8·41-s − 11·49-s + 12·59-s − 6·61-s + 24·71-s − 16·79-s + 81-s + 32·89-s + 12·99-s + 16·101-s − 14·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 3.61·11-s − 0.458·19-s + 2.22·29-s − 1.07·31-s + 1.24·41-s − 1.57·49-s + 1.56·59-s − 0.768·61-s + 2.84·71-s − 1.80·79-s + 1/9·81-s + 3.39·89-s + 1.20·99-s + 1.59·101-s − 1.34·109-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.30·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.317541145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.317541145\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.251850449594290770023595522342, −8.216327510194061992988360882605, −7.69173816326860941358833451220, −7.61226632157604454082531512011, −7.10045582332478116726435216083, −6.63387266185230904261033029596, −6.27741211109226051045636000355, −5.85571954014255792015786742978, −5.32941339333530498454835308131, −5.31528355424822668119220824680, −4.79627571666541903189528960577, −4.57746334930610966332857877715, −4.02001639384621541705560289088, −3.20945375736697531201024546740, −3.15444996637308104280530847817, −2.65188150747992211289103483389, −2.15143293081397859888884334114, −2.01856566755962819780839909400, −0.817809241830656324683263937974, −0.39772422318481380045722269006,
0.39772422318481380045722269006, 0.817809241830656324683263937974, 2.01856566755962819780839909400, 2.15143293081397859888884334114, 2.65188150747992211289103483389, 3.15444996637308104280530847817, 3.20945375736697531201024546740, 4.02001639384621541705560289088, 4.57746334930610966332857877715, 4.79627571666541903189528960577, 5.31528355424822668119220824680, 5.32941339333530498454835308131, 5.85571954014255792015786742978, 6.27741211109226051045636000355, 6.63387266185230904261033029596, 7.10045582332478116726435216083, 7.61226632157604454082531512011, 7.69173816326860941358833451220, 8.216327510194061992988360882605, 8.251850449594290770023595522342
