| L(s) = 1 | − 2·3-s + 4-s + 9-s − 2·12-s − 4·13-s − 2·25-s + 2·27-s + 36-s − 6·37-s + 8·39-s + 2·43-s + 2·49-s − 4·52-s − 64-s + 4·75-s − 4·79-s − 4·81-s − 2·100-s + 2·108-s + 12·111-s − 4·117-s − 121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2·3-s + 4-s + 9-s − 2·12-s − 4·13-s − 2·25-s + 2·27-s + 36-s − 6·37-s + 8·39-s + 2·43-s + 2·49-s − 4·52-s − 64-s + 4·75-s − 4·79-s − 4·81-s − 2·100-s + 2·108-s + 12·111-s − 4·117-s − 121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01013342911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01013342911\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.96847999741343284157400054131, −6.82063803073525890354755588796, −6.74524469799799543321089464835, −6.24411238120174680504624130616, −5.89103082518088967774685349419, −5.71792187966344685354929758470, −5.62081852328069182155219350527, −5.57110298786454059128633441288, −5.27692678650420233682855018245, −5.12475843274100895489732061141, −4.87975432594584617538101537904, −4.50929015634977972270348151579, −4.37977951679326964130762169847, −4.27945497313572184677321842329, −3.94108365881081078480346490966, −3.30476698576486538965350602633, −3.22924350471453690882809661607, −3.10181688433294231286970377255, −2.55972255514505600877049159427, −2.42695428394286595579480048953, −2.11834542637602205600736471499, −1.94827883820875282394748837074, −1.63352120373808205776458497356, −1.00651245830536881827950861833, −0.06972353008464826317171877562,
0.06972353008464826317171877562, 1.00651245830536881827950861833, 1.63352120373808205776458497356, 1.94827883820875282394748837074, 2.11834542637602205600736471499, 2.42695428394286595579480048953, 2.55972255514505600877049159427, 3.10181688433294231286970377255, 3.22924350471453690882809661607, 3.30476698576486538965350602633, 3.94108365881081078480346490966, 4.27945497313572184677321842329, 4.37977951679326964130762169847, 4.50929015634977972270348151579, 4.87975432594584617538101537904, 5.12475843274100895489732061141, 5.27692678650420233682855018245, 5.57110298786454059128633441288, 5.62081852328069182155219350527, 5.71792187966344685354929758470, 5.89103082518088967774685349419, 6.24411238120174680504624130616, 6.74524469799799543321089464835, 6.82063803073525890354755588796, 6.96847999741343284157400054131
