| L(s) = 1 | + 2·3-s + 3·9-s + 18·11-s − 12·17-s − 10·25-s + 10·27-s + 36·33-s − 6·41-s + 30·43-s + 14·49-s − 24·51-s + 18·59-s + 42·67-s + 4·73-s − 20·75-s + 20·81-s − 72·89-s − 10·97-s + 54·99-s − 36·113-s + 169·121-s − 12·123-s + 127-s + 60·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 5.42·11-s − 2.91·17-s − 2·25-s + 1.92·27-s + 6.26·33-s − 0.937·41-s + 4.57·43-s + 2·49-s − 3.36·51-s + 2.34·59-s + 5.13·67-s + 0.468·73-s − 2.30·75-s + 20/9·81-s − 7.63·89-s − 1.01·97-s + 5.42·99-s − 3.38·113-s + 15.3·121-s − 1.08·123-s + 0.0887·127-s + 5.28·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.946431577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.946431577\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
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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
−6.83160611092981703113557860497, −6.74831441716752133609682809311, −6.57625797432621719292953057777, −6.56085874468274761400507717713, −6.43230693778441145620554582484, −5.95117971439054158791494401243, −5.59403037148366398078113934628, −5.53489714490957096318361889190, −5.41502627198971662207472655108, −4.69384375454561129067102536872, −4.33943386328514172338208420114, −4.31938108324707879166464812781, −4.28072865889084406838617323534, −3.83997206633136918725499204307, −3.81790531014624614918755930043, −3.62993230901450466904051278201, −3.61428016264488997907405308835, −2.54733783820364604819152799462, −2.46950336005498249771140739518, −2.46541427700679762506862713237, −2.22609879932618066474748082400, −1.37067479429201724708762190799, −1.34061555251091415807904713298, −1.28075714711927147277216650166, −0.57812065755926670835597167160,
0.57812065755926670835597167160, 1.28075714711927147277216650166, 1.34061555251091415807904713298, 1.37067479429201724708762190799, 2.22609879932618066474748082400, 2.46541427700679762506862713237, 2.46950336005498249771140739518, 2.54733783820364604819152799462, 3.61428016264488997907405308835, 3.62993230901450466904051278201, 3.81790531014624614918755930043, 3.83997206633136918725499204307, 4.28072865889084406838617323534, 4.31938108324707879166464812781, 4.33943386328514172338208420114, 4.69384375454561129067102536872, 5.41502627198971662207472655108, 5.53489714490957096318361889190, 5.59403037148366398078113934628, 5.95117971439054158791494401243, 6.43230693778441145620554582484, 6.56085874468274761400507717713, 6.57625797432621719292953057777, 6.74831441716752133609682809311, 6.83160611092981703113557860497
