| L(s) = 1 | − 2·5-s − 2·7-s + 2·11-s + 6·13-s + 4·17-s − 6·23-s + 3·25-s − 10·29-s + 4·31-s + 4·35-s + 4·37-s + 2·41-s + 6·43-s − 4·47-s + 2·49-s + 4·53-s − 4·55-s − 10·59-s + 6·61-s − 12·65-s + 16·67-s + 22·71-s + 18·73-s − 4·77-s + 16·79-s + 6·83-s − 8·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 0.603·11-s + 1.66·13-s + 0.970·17-s − 1.25·23-s + 3/5·25-s − 1.85·29-s + 0.718·31-s + 0.676·35-s + 0.657·37-s + 0.312·41-s + 0.914·43-s − 0.583·47-s + 2/7·49-s + 0.549·53-s − 0.539·55-s − 1.30·59-s + 0.768·61-s − 1.48·65-s + 1.95·67-s + 2.61·71-s + 2.10·73-s − 0.455·77-s + 1.80·79-s + 0.658·83-s − 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.116745126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.116745126\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 53 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 97 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 22 T + 250 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 209 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.373616951211482931305486930683, −8.860625114534061401680084711761, −8.356920610328689021621545649650, −8.155686972423249065956332214375, −7.73145558930634095324664582829, −7.50792801773580936595719583114, −6.84440747467569559318715572887, −6.43275139509929433468228465660, −6.27050832250318788228252721410, −5.81033899105642342271690080626, −5.32625661129792113293438488851, −4.90464937006232093368101117830, −4.12335232822114281437710012430, −3.87032773014820156685160600053, −3.52226796508639220118477455658, −3.37468159017292957538864731174, −2.41039243861264022261278171164, −1.98243830582799593825103542093, −1.05144717746452817314141476217, −0.62407517810417529749798059963,
0.62407517810417529749798059963, 1.05144717746452817314141476217, 1.98243830582799593825103542093, 2.41039243861264022261278171164, 3.37468159017292957538864731174, 3.52226796508639220118477455658, 3.87032773014820156685160600053, 4.12335232822114281437710012430, 4.90464937006232093368101117830, 5.32625661129792113293438488851, 5.81033899105642342271690080626, 6.27050832250318788228252721410, 6.43275139509929433468228465660, 6.84440747467569559318715572887, 7.50792801773580936595719583114, 7.73145558930634095324664582829, 8.155686972423249065956332214375, 8.356920610328689021621545649650, 8.860625114534061401680084711761, 9.373616951211482931305486930683
