| L(s) = 1 | − 2·3-s + 2·4-s − 2·5-s + 3·9-s − 2·11-s − 4·12-s − 4·13-s + 4·15-s + 6·17-s + 2·19-s − 4·20-s + 6·23-s + 3·25-s − 4·27-s − 6·29-s − 4·31-s + 4·33-s + 6·36-s + 4·37-s + 8·39-s + 10·43-s − 4·44-s − 6·45-s + 12·47-s − 12·51-s − 8·52-s + 6·53-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 4-s − 0.894·5-s + 9-s − 0.603·11-s − 1.15·12-s − 1.10·13-s + 1.03·15-s + 1.45·17-s + 0.458·19-s − 0.894·20-s + 1.25·23-s + 3/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.696·33-s + 36-s + 0.657·37-s + 1.28·39-s + 1.52·43-s − 0.603·44-s − 0.894·45-s + 1.75·47-s − 1.68·51-s − 1.10·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65367225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65367225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.804818937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804818937\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 61 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 105 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 121 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 147 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 120 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 133 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 189 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.63317542108191133250292239455, −7.57667385656903148792222041647, −7.33684668391861621135557773417, −6.98362656456048549236813217679, −6.85241537629348982063734664837, −6.21626946857673487408397875078, −5.82065968194901159332753915823, −5.59537136000910437357915265020, −5.27967112206468204806337846022, −5.03863325924604212795179440461, −4.32739604790202562031496839810, −4.31943274350839641952052127864, −3.73140038945239688719485620181, −3.20192274866353480932133324721, −2.96037248373155308930258126221, −2.37436132395242052616215941616, −2.11804480168481961061885260530, −1.37348942201757540772843226264, −0.847574128882473204358542605538, −0.43854059735775056049974220991,
0.43854059735775056049974220991, 0.847574128882473204358542605538, 1.37348942201757540772843226264, 2.11804480168481961061885260530, 2.37436132395242052616215941616, 2.96037248373155308930258126221, 3.20192274866353480932133324721, 3.73140038945239688719485620181, 4.31943274350839641952052127864, 4.32739604790202562031496839810, 5.03863325924604212795179440461, 5.27967112206468204806337846022, 5.59537136000910437357915265020, 5.82065968194901159332753915823, 6.21626946857673487408397875078, 6.85241537629348982063734664837, 6.98362656456048549236813217679, 7.33684668391861621135557773417, 7.57667385656903148792222041647, 7.63317542108191133250292239455
