| L(s) = 1 | + 4·19-s − 4·41-s + 8·59-s + 81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 4·19-s − 4·41-s + 8·59-s + 81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.445783740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.445783740\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$ | \( ( 1 - T )^{8} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
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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
−7.06903753432400722442128225464, −6.83801105213854819093665737452, −6.45554086269102033428833572921, −6.35404777405152463038642474250, −6.11209615441944494090003650544, −5.56915779428869497575206393200, −5.53665741941759679655218287337, −5.40027115487692109183849460081, −5.27901150502888483240493463928, −4.93844714229888410008396902722, −4.92850778866561085010260028843, −4.69228268137439700874254290352, −4.05209147082459860534063871823, −3.86688181758069073815050991044, −3.76692036608279088586439958730, −3.61498629578203627218172465436, −3.20947049209898691190974869480, −3.04929714164307614392700433941, −2.92308736652548572316341251537, −2.30682283046669900738831971345, −2.09494165064015195096684179690, −2.07842646331374109940518643082, −1.24259511963366857286333118685, −1.17002212834116417923128429689, −0.854792395949630825824052265300,
0.854792395949630825824052265300, 1.17002212834116417923128429689, 1.24259511963366857286333118685, 2.07842646331374109940518643082, 2.09494165064015195096684179690, 2.30682283046669900738831971345, 2.92308736652548572316341251537, 3.04929714164307614392700433941, 3.20947049209898691190974869480, 3.61498629578203627218172465436, 3.76692036608279088586439958730, 3.86688181758069073815050991044, 4.05209147082459860534063871823, 4.69228268137439700874254290352, 4.92850778866561085010260028843, 4.93844714229888410008396902722, 5.27901150502888483240493463928, 5.40027115487692109183849460081, 5.53665741941759679655218287337, 5.56915779428869497575206393200, 6.11209615441944494090003650544, 6.35404777405152463038642474250, 6.45554086269102033428833572921, 6.83801105213854819093665737452, 7.06903753432400722442128225464
