| L(s) = 1 | − 2·7-s − 3·13-s − 3·19-s − 25-s − 37-s − 43-s + 3·49-s − 2·67-s + 3·73-s − 2·79-s + 6·91-s − 3·103-s − 109-s + 121-s + 127-s + 131-s + 6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s + 2·175-s + ⋯ |
| L(s) = 1 | − 2·7-s − 3·13-s − 3·19-s − 25-s − 37-s − 43-s + 3·49-s − 2·67-s + 3·73-s − 2·79-s + 6·91-s − 3·103-s − 109-s + 121-s + 127-s + 131-s + 6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s + 2·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1041229574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1041229574\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−9.574204820740111496243966049823, −9.018151897168569056718004338847, −8.819159151122178194926460403699, −8.075151984491076032076500333442, −8.024770078571018137125466061662, −7.30208972662518895185062778617, −7.00081280337659589689183162227, −6.61833868672080014786231068553, −6.58279203004158094002444953168, −5.77702885276373498323598161659, −5.67479181888446398596094503121, −4.92982020567103193352330236177, −4.64136921014109182481151666043, −3.98683951580907400592450114206, −3.90316651102538969112221399413, −2.96457243555450345273859150733, −2.80711424125016722309785520725, −2.17775552206390540664676417279, −1.89191983440623025157436022048, −0.19936798300695318836333084695,
0.19936798300695318836333084695, 1.89191983440623025157436022048, 2.17775552206390540664676417279, 2.80711424125016722309785520725, 2.96457243555450345273859150733, 3.90316651102538969112221399413, 3.98683951580907400592450114206, 4.64136921014109182481151666043, 4.92982020567103193352330236177, 5.67479181888446398596094503121, 5.77702885276373498323598161659, 6.58279203004158094002444953168, 6.61833868672080014786231068553, 7.00081280337659589689183162227, 7.30208972662518895185062778617, 8.024770078571018137125466061662, 8.075151984491076032076500333442, 8.819159151122178194926460403699, 9.018151897168569056718004338847, 9.574204820740111496243966049823
