| L(s) = 1 | + 4·3-s − 4·7-s + 6·9-s + 8·13-s − 16-s + 4·19-s − 16·21-s − 4·27-s − 20·31-s − 4·37-s + 32·39-s − 4·48-s + 8·49-s + 16·57-s + 32·61-s − 24·63-s + 20·67-s − 4·73-s − 40·79-s − 37·81-s − 32·91-s − 80·93-s − 28·97-s + 4·109-s − 16·111-s + 4·112-s + 48·117-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.51·7-s + 2·9-s + 2.21·13-s − 1/4·16-s + 0.917·19-s − 3.49·21-s − 0.769·27-s − 3.59·31-s − 0.657·37-s + 5.12·39-s − 0.577·48-s + 8/7·49-s + 2.11·57-s + 4.09·61-s − 3.02·63-s + 2.44·67-s − 0.468·73-s − 4.50·79-s − 4.11·81-s − 3.35·91-s − 8.29·93-s − 2.84·97-s + 0.383·109-s − 1.51·111-s + 0.377·112-s + 4.43·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.677004440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.677004440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 + T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 2722 T^{4} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 1666 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 3442 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 + 5794 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 3374 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 15518 T^{4} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.22962409609456433245768818249, −7.07016635259605825682806147076, −6.86915256857119081157305746333, −6.55671272987272668850243569332, −6.28587054128424301410234678989, −6.11303450444885643996722076195, −5.64641845260248698870777580206, −5.58271078423323100671504545178, −5.51681358829249898529448746781, −5.26020901751273711743845626171, −4.87486010028116000615906200460, −4.30153597556576245932011039173, −4.10649225898848003879516446271, −3.77028440778335171036997422539, −3.73149944900423468378206603820, −3.54352803548533372590464931605, −3.42587723280838639040613155157, −3.10134724492919741928496646964, −2.61279009361964217256241870082, −2.51197808784028607522587919343, −2.35446875000867515281987056864, −1.69219593981253661912035641300, −1.55626953031201905766173472793, −1.15692906186722819687092818764, −0.29505375134976694326419730739,
0.29505375134976694326419730739, 1.15692906186722819687092818764, 1.55626953031201905766173472793, 1.69219593981253661912035641300, 2.35446875000867515281987056864, 2.51197808784028607522587919343, 2.61279009361964217256241870082, 3.10134724492919741928496646964, 3.42587723280838639040613155157, 3.54352803548533372590464931605, 3.73149944900423468378206603820, 3.77028440778335171036997422539, 4.10649225898848003879516446271, 4.30153597556576245932011039173, 4.87486010028116000615906200460, 5.26020901751273711743845626171, 5.51681358829249898529448746781, 5.58271078423323100671504545178, 5.64641845260248698870777580206, 6.11303450444885643996722076195, 6.28587054128424301410234678989, 6.55671272987272668850243569332, 6.86915256857119081157305746333, 7.07016635259605825682806147076, 7.22962409609456433245768818249
