| L(s) = 1 | + 2·3-s + 3·9-s + 2·13-s − 4·19-s + 25-s + 4·27-s + 4·39-s + 2·49-s − 8·57-s + 2·75-s − 2·79-s + 5·81-s − 2·97-s + 2·103-s + 6·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯ |
| L(s) = 1 | + 2·3-s + 3·9-s + 2·13-s − 4·19-s + 25-s + 4·27-s + 4·39-s + 2·49-s − 8·57-s + 2·75-s − 2·79-s + 5·81-s − 2·97-s + 2·103-s + 6·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14107536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14107536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.580459730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.580459730\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 313 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.785851370520609655250561535334, −8.378741355072920119261002382737, −8.290814093835749620584621269983, −8.211717875853318985983843868172, −7.37170171053222653955698977228, −7.07760382964634302775289750309, −6.79700923728156894971029763593, −6.37742232697386040078138806974, −6.00451322326766499426072678000, −5.71273002538015404164899343797, −4.74602307009875378489814719921, −4.52169411030607859650399790657, −4.14219077583304853849486602761, −3.86300676578161777553435165842, −3.47866879129007037207351525506, −2.94541026064580168649098360662, −2.41197936041089706818323620496, −2.14323717241113469601512258149, −1.61004259175774456264277347381, −1.03599310237343597690680986044,
1.03599310237343597690680986044, 1.61004259175774456264277347381, 2.14323717241113469601512258149, 2.41197936041089706818323620496, 2.94541026064580168649098360662, 3.47866879129007037207351525506, 3.86300676578161777553435165842, 4.14219077583304853849486602761, 4.52169411030607859650399790657, 4.74602307009875378489814719921, 5.71273002538015404164899343797, 6.00451322326766499426072678000, 6.37742232697386040078138806974, 6.79700923728156894971029763593, 7.07760382964634302775289750309, 7.37170171053222653955698977228, 8.211717875853318985983843868172, 8.290814093835749620584621269983, 8.378741355072920119261002382737, 8.785851370520609655250561535334
