L(s) = 1 | − 3·3-s − 4·4-s + 4·5-s − 14·7-s + 6·9-s − 18·11-s + 12·12-s + 13·13-s − 12·15-s + 12·17-s + 14·19-s − 16·20-s + 42·21-s − 22·23-s − 38·25-s − 9·27-s + 56·28-s − 34·29-s + 14·31-s + 54·33-s − 56·35-s − 24·36-s + 12·37-s − 39·39-s − 56·41-s − 65·43-s + 72·44-s + ⋯ |
L(s) = 1 | − 3-s − 4-s + 4/5·5-s − 2·7-s + 2/3·9-s − 1.63·11-s + 12-s + 13-s − 4/5·15-s + 0.705·17-s + 0.736·19-s − 4/5·20-s + 2·21-s − 0.956·23-s − 1.51·25-s − 1/3·27-s + 2·28-s − 1.17·29-s + 0.451·31-s + 1.63·33-s − 8/5·35-s − 2/3·36-s + 0.324·37-s − 39-s − 1.36·41-s − 1.51·43-s + 1.63·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03238829623\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03238829623\) |
\(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 18 T + 229 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 12 T + 337 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T - 165 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 22 T - 45 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 34 T + 315 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 12 T + 1417 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 56 T + 1455 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 65 T + 2376 T^{2} + 65 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 62 T + 363 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 111 T + 7828 T^{2} - 111 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 21 T + 4636 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 114 T + 9373 T^{2} - 114 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 19 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 64 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 7905 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + T - 9408 T^{2} + p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.95750541206043890256308514185, −11.44573125022304634614157349044, −10.96732505950600749834071714562, −10.16325089547673071462183516143, −9.975129731690030203924633871428, −9.699156903439851732834741222450, −9.475952536547470380280841738293, −8.470519671421501974423354795438, −8.188873116776310557061142028453, −7.49071575902603882800821290668, −6.76214211730971821105735697678, −6.25632090736526185939702483882, −5.92354225942949246293516807115, −5.21708144835507501150910254417, −5.17280155476593777355406962919, −3.93732554339697540155751688010, −3.56891135014607223275459845875, −2.77914872701230453795294111248, −1.63920309903169203477519071164, −0.095904846523508496938699133562,
0.095904846523508496938699133562, 1.63920309903169203477519071164, 2.77914872701230453795294111248, 3.56891135014607223275459845875, 3.93732554339697540155751688010, 5.17280155476593777355406962919, 5.21708144835507501150910254417, 5.92354225942949246293516807115, 6.25632090736526185939702483882, 6.76214211730971821105735697678, 7.49071575902603882800821290668, 8.188873116776310557061142028453, 8.470519671421501974423354795438, 9.475952536547470380280841738293, 9.699156903439851732834741222450, 9.975129731690030203924633871428, 10.16325089547673071462183516143, 10.96732505950600749834071714562, 11.44573125022304634614157349044, 11.95750541206043890256308514185