| L(s) = 1 | − 4·13-s − 4·25-s + 4·49-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-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 + 241-s + ⋯ |
| L(s) = 1 | − 4·13-s − 4·25-s + 4·49-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-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 + 241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4333698967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4333698967\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−6.14195323990207397753923975105, −6.02057743765917445472065624291, −5.74070716771980778097369678175, −5.55145228863294958785239689606, −5.43394498138538793666014449607, −5.27149342614370267855257778935, −5.05560083039814757990789237622, −4.71369326503036973267243960281, −4.65401659035196692313075182019, −4.53082881434973993644841442036, −4.09611825033126590191091637393, −3.92122169681425779021643531621, −3.88568279264703696013386752839, −3.70010525925865396020261816861, −3.34939377710441144240597215543, −2.95335226492276597086962578365, −2.67584823095685293538685839265, −2.61746241635759860519042177421, −2.20691513524190528126646672082, −2.19245372625571115517203343922, −2.16877530925062972290686873833, −1.57446386799986367762378332276, −1.40632494557808042055630762595, −0.75204837644625229371663867573, −0.28014224669104248341316713065,
0.28014224669104248341316713065, 0.75204837644625229371663867573, 1.40632494557808042055630762595, 1.57446386799986367762378332276, 2.16877530925062972290686873833, 2.19245372625571115517203343922, 2.20691513524190528126646672082, 2.61746241635759860519042177421, 2.67584823095685293538685839265, 2.95335226492276597086962578365, 3.34939377710441144240597215543, 3.70010525925865396020261816861, 3.88568279264703696013386752839, 3.92122169681425779021643531621, 4.09611825033126590191091637393, 4.53082881434973993644841442036, 4.65401659035196692313075182019, 4.71369326503036973267243960281, 5.05560083039814757990789237622, 5.27149342614370267855257778935, 5.43394498138538793666014449607, 5.55145228863294958785239689606, 5.74070716771980778097369678175, 6.02057743765917445472065624291, 6.14195323990207397753923975105
