| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s + 3·9-s + 2·11-s − 2·13-s + 4·15-s + 4·17-s + 6·19-s − 4·21-s + 3·25-s + 4·27-s + 4·29-s − 2·31-s + 4·33-s − 4·35-s + 4·37-s − 4·39-s + 4·41-s + 4·43-s + 6·45-s + 3·49-s + 8·51-s − 2·53-s + 4·55-s + 12·57-s − 8·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s + 0.603·11-s − 0.554·13-s + 1.03·15-s + 0.970·17-s + 1.37·19-s − 0.872·21-s + 3/5·25-s + 0.769·27-s + 0.742·29-s − 0.359·31-s + 0.696·33-s − 0.676·35-s + 0.657·37-s − 0.640·39-s + 0.624·41-s + 0.609·43-s + 0.894·45-s + 3/7·49-s + 1.12·51-s − 0.274·53-s + 0.539·55-s + 1.58·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.783343983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.783343983\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 2 T - 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 22 T + 250 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 258 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−8.058543574427335626805023716202, −7.85257014440105896196143200969, −7.40686209308061927365299567961, −7.26697667169170124657974752850, −6.74795038615466565766836338760, −6.44928105533423276853415225053, −6.12689826002008429221544980351, −5.72612488907770516193443106055, −5.17414670875941747052104153232, −5.16146886876554866313930293945, −4.47849591494988199274333261964, −4.09723997579746101833295523469, −3.67790014466478805281852012189, −3.28642970886119721594219896259, −2.81473115612901459576922324511, −2.78440747194449912846437674823, −2.01792068801978651513931792596, −1.77039458467496719640609407328, −0.978628911983950495583869841343, −0.75264118017419226924907195799,
0.75264118017419226924907195799, 0.978628911983950495583869841343, 1.77039458467496719640609407328, 2.01792068801978651513931792596, 2.78440747194449912846437674823, 2.81473115612901459576922324511, 3.28642970886119721594219896259, 3.67790014466478805281852012189, 4.09723997579746101833295523469, 4.47849591494988199274333261964, 5.16146886876554866313930293945, 5.17414670875941747052104153232, 5.72612488907770516193443106055, 6.12689826002008429221544980351, 6.44928105533423276853415225053, 6.74795038615466565766836338760, 7.26697667169170124657974752850, 7.40686209308061927365299567961, 7.85257014440105896196143200969, 8.058543574427335626805023716202
