| L(s) = 1 | − 2·5-s − 10·7-s + 8·11-s − 5·13-s + 8·19-s − 6·23-s + 5·25-s + 8·29-s − 2·31-s + 20·35-s + 14·37-s + 11·43-s + 10·47-s + 61·49-s + 6·53-s − 16·55-s + 8·59-s + 61-s + 10·65-s − 5·67-s + 6·71-s − 73-s − 80·77-s + 13·79-s + 8·83-s − 12·89-s + 50·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 3.77·7-s + 2.41·11-s − 1.38·13-s + 1.83·19-s − 1.25·23-s + 25-s + 1.48·29-s − 0.359·31-s + 3.38·35-s + 2.30·37-s + 1.67·43-s + 1.45·47-s + 61/7·49-s + 0.824·53-s − 2.15·55-s + 1.04·59-s + 0.128·61-s + 1.24·65-s − 0.610·67-s + 0.712·71-s − 0.117·73-s − 9.11·77-s + 1.46·79-s + 0.878·83-s − 1.27·89-s + 5.24·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.202068945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.202068945\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.531370913678215225987514938255, −9.511817834063076143908708545735, −9.152765804842676781300220776966, −8.979080270217750820889251497712, −8.056451897185895959503858654128, −7.70642857849225812148347453234, −7.06048864473501021378274097888, −6.97476743059609720446613826082, −6.44452259679676315810798021669, −6.43964947415534288960490186858, −5.60798296581914476499247281099, −5.55204730071256908803073949132, −4.23503813045670250060714363338, −4.17194870668169925522015491466, −3.81163162798422794495124726086, −3.24556666742681095049391014653, −2.75201122152921653874782134088, −2.54828314860183541123936751157, −0.929674237972465586620088986282, −0.60765367554852474063342895806,
0.60765367554852474063342895806, 0.929674237972465586620088986282, 2.54828314860183541123936751157, 2.75201122152921653874782134088, 3.24556666742681095049391014653, 3.81163162798422794495124726086, 4.17194870668169925522015491466, 4.23503813045670250060714363338, 5.55204730071256908803073949132, 5.60798296581914476499247281099, 6.43964947415534288960490186858, 6.44452259679676315810798021669, 6.97476743059609720446613826082, 7.06048864473501021378274097888, 7.70642857849225812148347453234, 8.056451897185895959503858654128, 8.979080270217750820889251497712, 9.152765804842676781300220776966, 9.511817834063076143908708545735, 9.531370913678215225987514938255
