| L(s) = 1 | − 2·3-s − 6·11-s − 4·13-s + 2·17-s − 4·19-s − 6·23-s + 2·25-s + 2·27-s + 2·31-s + 12·33-s + 16·37-s + 8·39-s + 12·41-s + 4·43-s − 4·51-s + 12·53-s + 8·57-s − 12·59-s + 8·61-s + 16·67-s + 12·69-s − 6·71-s − 4·73-s − 4·75-s − 14·79-s − 81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.80·11-s − 1.10·13-s + 0.485·17-s − 0.917·19-s − 1.25·23-s + 2/5·25-s + 0.384·27-s + 0.359·31-s + 2.08·33-s + 2.63·37-s + 1.28·39-s + 1.87·41-s + 0.609·43-s − 0.560·51-s + 1.64·53-s + 1.05·57-s − 1.56·59-s + 1.02·61-s + 1.95·67-s + 1.44·69-s − 0.712·71-s − 0.468·73-s − 0.461·75-s − 1.57·79-s − 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9941565531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9941565531\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 180 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 202 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.818153318459398069101157941562, −8.313125117731867555096215164111, −7.953397023749231507545524513756, −7.57911304538136716773356308706, −7.56591157354505723402201413587, −6.94550373743976103990537032774, −6.33618070971745890198037535139, −6.18264218925600596420924077964, −5.65507509522548050914422598260, −5.53225517629595637057961942293, −5.17592102467986971930883989365, −4.52798599579234357266094980565, −4.34448104068425850990046901692, −3.98261627505997134413863762394, −3.01451360060443167709472917403, −2.77423330181519051161112403393, −2.34354982380820443903163491900, −1.91829352062359954694482555732, −0.70815586165099889238506353727, −0.49665332181083757423666218790,
0.49665332181083757423666218790, 0.70815586165099889238506353727, 1.91829352062359954694482555732, 2.34354982380820443903163491900, 2.77423330181519051161112403393, 3.01451360060443167709472917403, 3.98261627505997134413863762394, 4.34448104068425850990046901692, 4.52798599579234357266094980565, 5.17592102467986971930883989365, 5.53225517629595637057961942293, 5.65507509522548050914422598260, 6.18264218925600596420924077964, 6.33618070971745890198037535139, 6.94550373743976103990537032774, 7.56591157354505723402201413587, 7.57911304538136716773356308706, 7.953397023749231507545524513756, 8.313125117731867555096215164111, 8.818153318459398069101157941562
