| L(s) = 1 | + 2·5-s + 7-s − 11-s + 6·13-s + 5·17-s + 3·19-s − 7·23-s + 3·25-s + 29-s + 13·31-s + 2·35-s + 7·37-s + 2·41-s + 5·43-s − 3·47-s + 5·49-s + 2·53-s − 2·55-s − 24·59-s − 7·61-s + 12·65-s + 21·67-s − 10·71-s + 73-s − 77-s − 10·79-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 0.301·11-s + 1.66·13-s + 1.21·17-s + 0.688·19-s − 1.45·23-s + 3/5·25-s + 0.185·29-s + 2.33·31-s + 0.338·35-s + 1.15·37-s + 0.312·41-s + 0.762·43-s − 0.437·47-s + 5/7·49-s + 0.274·53-s − 0.269·55-s − 3.12·59-s − 0.896·61-s + 1.48·65-s + 2.56·67-s − 1.18·71-s + 0.117·73-s − 0.113·77-s − 1.12·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.409201386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.409201386\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 40 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 13 T + 86 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 68 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 74 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 78 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 116 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 21 T + 226 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 128 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−10.06950417882576880808886772412, −9.881342831571658890458184278689, −9.244514228403282209991745455007, −8.871396403497008980934232853459, −8.359516381603993741077479895122, −8.138760321060995257829332127947, −7.49836179084295472784312208454, −7.46156186594652184006741341198, −6.37892643896390471232880589886, −6.29988298884589201728014885159, −5.78984918927165361092251625453, −5.68017052219723964540283539195, −4.78669483850413394177267422832, −4.53806753810589408016431165344, −3.90977358029582823112183821314, −3.25460851784024562621542818478, −2.85201551697768493989389785198, −2.16390211514549087645485351507, −1.36682501051982937142302674576, −0.962732603772212041364135582889,
0.962732603772212041364135582889, 1.36682501051982937142302674576, 2.16390211514549087645485351507, 2.85201551697768493989389785198, 3.25460851784024562621542818478, 3.90977358029582823112183821314, 4.53806753810589408016431165344, 4.78669483850413394177267422832, 5.68017052219723964540283539195, 5.78984918927165361092251625453, 6.29988298884589201728014885159, 6.37892643896390471232880589886, 7.46156186594652184006741341198, 7.49836179084295472784312208454, 8.138760321060995257829332127947, 8.359516381603993741077479895122, 8.871396403497008980934232853459, 9.244514228403282209991745455007, 9.881342831571658890458184278689, 10.06950417882576880808886772412
