| L(s) = 1 | + 4-s − 5·9-s − 14·19-s + 20·29-s − 8·31-s − 5·36-s − 4·41-s + 20·49-s − 2·59-s − 16·61-s − 64-s + 24·71-s − 14·76-s − 8·79-s + 9·81-s − 26·89-s − 4·101-s − 4·109-s + 20·116-s − 44·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 5/3·9-s − 3.21·19-s + 3.71·29-s − 1.43·31-s − 5/6·36-s − 0.624·41-s + 20/7·49-s − 0.260·59-s − 2.04·61-s − 1/8·64-s + 2.84·71-s − 1.60·76-s − 0.900·79-s + 81-s − 2.75·89-s − 0.398·101-s − 0.383·109-s + 1.85·116-s − 4·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07831924092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07831924092\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2^3$ | \( 1 - 15 T^{2} - 64 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 58 T^{2} + 1515 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 137 T^{2} + 13440 T^{4} + 137 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - 31 T^{2} - 8448 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.24903724510044528233062592458, −7.01161453847289977576748745334, −6.63303956935156635565896586996, −6.35787579211384244387026013447, −6.28187217984361891906604832587, −6.26383761212878209321051787158, −5.99762977822599129886135023561, −5.51493464702251269920713037866, −5.40417115485968476051934839357, −5.19950837468394501833989652945, −4.92186111585557371088124051254, −4.51843792320237024018355204104, −4.43416090502937850857314655864, −4.07785054821255127927936154037, −3.80482947237208209895949919534, −3.77849189061787920699576941870, −3.02572344753702151810622451150, −3.00779027933957581370270984600, −2.62391822363794850042053879788, −2.53810334306240301171549339700, −2.16012183911447001980077903740, −1.92439053178790940389031801423, −1.29976587523019975804123353436, −0.978587380227841306439929694807, −0.06793423767732642485539334538,
0.06793423767732642485539334538, 0.978587380227841306439929694807, 1.29976587523019975804123353436, 1.92439053178790940389031801423, 2.16012183911447001980077903740, 2.53810334306240301171549339700, 2.62391822363794850042053879788, 3.00779027933957581370270984600, 3.02572344753702151810622451150, 3.77849189061787920699576941870, 3.80482947237208209895949919534, 4.07785054821255127927936154037, 4.43416090502937850857314655864, 4.51843792320237024018355204104, 4.92186111585557371088124051254, 5.19950837468394501833989652945, 5.40417115485968476051934839357, 5.51493464702251269920713037866, 5.99762977822599129886135023561, 6.26383761212878209321051787158, 6.28187217984361891906604832587, 6.35787579211384244387026013447, 6.63303956935156635565896586996, 7.01161453847289977576748745334, 7.24903724510044528233062592458
