| L(s) = 1 | + 2-s + 3-s + 6-s − 8-s + 11-s − 16-s + 2·17-s + 4·19-s + 22-s − 24-s − 27-s + 33-s + 2·34-s + 4·38-s − 2·41-s − 43-s − 48-s − 49-s + 2·51-s − 54-s + 4·57-s + 59-s + 64-s + 66-s + 2·67-s − 4·73-s − 81-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 6-s − 8-s + 11-s − 16-s + 2·17-s + 4·19-s + 22-s − 24-s − 27-s + 33-s + 2·34-s + 4·38-s − 2·41-s − 43-s − 48-s − 49-s + 2·51-s − 54-s + 4·57-s + 59-s + 64-s + 66-s + 2·67-s − 4·73-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.827677596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.827677596\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.625680110464385798226153276670, −9.471096139337727433943782610534, −8.829115645374614438163385918510, −8.465221021293952264873400573034, −8.122281476155593061212566328679, −7.75427263841609384953768013927, −7.22642292379932214219448988125, −6.96916159648424128705777112283, −6.52532524697068355893826548040, −5.63338561099498685332799190520, −5.53738507166212527556685598674, −5.39525608326456279268509654776, −4.81099938687918431875492147447, −4.12925643737480546430311282478, −3.59544207078539963036210579025, −3.42510382577416124173511658769, −2.94069395113731444155815619756, −2.80555211681645716994956358541, −1.43876718605282742789361477591, −1.32656648343528906529127900678,
1.32656648343528906529127900678, 1.43876718605282742789361477591, 2.80555211681645716994956358541, 2.94069395113731444155815619756, 3.42510382577416124173511658769, 3.59544207078539963036210579025, 4.12925643737480546430311282478, 4.81099938687918431875492147447, 5.39525608326456279268509654776, 5.53738507166212527556685598674, 5.63338561099498685332799190520, 6.52532524697068355893826548040, 6.96916159648424128705777112283, 7.22642292379932214219448988125, 7.75427263841609384953768013927, 8.122281476155593061212566328679, 8.465221021293952264873400573034, 8.829115645374614438163385918510, 9.471096139337727433943782610534, 9.625680110464385798226153276670
