| L(s) = 1 | + 4-s + 9-s + 10·11-s − 4·19-s + 4·29-s − 8·31-s + 36-s − 20·41-s + 10·44-s + 2·49-s + 22·61-s − 64-s − 6·71-s − 4·76-s + 56·79-s − 20·89-s + 10·99-s − 16·101-s + 68·109-s + 4·116-s + 47·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s + 3.01·11-s − 0.917·19-s + 0.742·29-s − 1.43·31-s + 1/6·36-s − 3.12·41-s + 1.50·44-s + 2/7·49-s + 2.81·61-s − 1/8·64-s − 0.712·71-s − 0.458·76-s + 6.30·79-s − 2.11·89-s + 1.00·99-s − 1.59·101-s + 6.51·109-s + 0.371·116-s + 4.27·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.428032125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.428032125\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 3 T^{2} - 520 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 157 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 25 T^{2} - 8784 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 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
−6.50674127507435493388120674563, −6.36636384229009668611591100939, −6.23497473850791954699406539573, −5.96209496310190102242585348402, −5.90623772704697366157656126396, −5.40432361788378295652217242739, −5.10365632238394575865161866739, −5.08777163963666586914039413751, −4.97868449131933576665692568070, −4.50478007217897186720030307842, −4.30546948913940443878119434557, −4.02320849731883608034976185900, −3.88114173349693897532228854474, −3.81211176272573082227189862131, −3.35103532621035465338961479715, −3.25792131650091172032139843810, −3.14959504062099845945697567533, −2.61410593988634643925066812758, −2.13931431218223168429903542002, −1.97218003112051893572499059602, −1.85059122588452714682700881461, −1.67745040428448953757987391605, −1.15353378342077417082246392123, −0.69366778245705257517750991252, −0.58433701139004941728636040394,
0.58433701139004941728636040394, 0.69366778245705257517750991252, 1.15353378342077417082246392123, 1.67745040428448953757987391605, 1.85059122588452714682700881461, 1.97218003112051893572499059602, 2.13931431218223168429903542002, 2.61410593988634643925066812758, 3.14959504062099845945697567533, 3.25792131650091172032139843810, 3.35103532621035465338961479715, 3.81211176272573082227189862131, 3.88114173349693897532228854474, 4.02320849731883608034976185900, 4.30546948913940443878119434557, 4.50478007217897186720030307842, 4.97868449131933576665692568070, 5.08777163963666586914039413751, 5.10365632238394575865161866739, 5.40432361788378295652217242739, 5.90623772704697366157656126396, 5.96209496310190102242585348402, 6.23497473850791954699406539573, 6.36636384229009668611591100939, 6.50674127507435493388120674563
