| L(s) = 1 | + 4-s + 2·5-s + 9-s + 4·19-s + 2·20-s + 5·25-s + 32·29-s − 8·31-s + 36-s + 40·41-s + 2·45-s + 24·59-s − 4·61-s − 64-s + 24·71-s + 4·76-s + 8·79-s − 28·89-s + 8·95-s + 5·100-s + 12·101-s − 4·109-s + 32·116-s + 22·121-s − 8·124-s + 22·125-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.894·5-s + 1/3·9-s + 0.917·19-s + 0.447·20-s + 25-s + 5.94·29-s − 1.43·31-s + 1/6·36-s + 6.24·41-s + 0.298·45-s + 3.12·59-s − 0.512·61-s − 1/8·64-s + 2.84·71-s + 0.458·76-s + 0.900·79-s − 2.96·89-s + 0.820·95-s + 1/2·100-s + 1.19·101-s − 0.383·109-s + 2.97·116-s + 2·121-s − 0.718·124-s + 1.96·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.24883130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.24883130\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 18 T^{2} - 205 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 38 T^{2} + 75 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.80034222082921771865042816481, −6.42361391667562408307510778561, −6.23820042770512513241332509429, −6.22700092173032153610153034138, −6.16780896932967141824504476078, −5.69811465275380996041629129026, −5.36293973070893895827501179779, −5.30729381166624692909130761414, −5.13266709729022515351766736213, −4.77539691794676714674716141375, −4.58911067733755502382119721115, −4.31908122275157766177104608443, −4.03128747738821136445894920105, −4.02646727411456748207857783367, −3.56594789470210554364709677015, −3.21636513982893502262985115531, −2.82519728467552799895120944827, −2.78782435644693789294669359927, −2.43369053393945337683509183049, −2.34149241273151373401002208591, −2.17096295884122059488048914414, −1.39957717309160398726180073344, −1.07840116367716814702412847862, −0.886516210025736502246084530831, −0.78514812779563027885484686310,
0.78514812779563027885484686310, 0.886516210025736502246084530831, 1.07840116367716814702412847862, 1.39957717309160398726180073344, 2.17096295884122059488048914414, 2.34149241273151373401002208591, 2.43369053393945337683509183049, 2.78782435644693789294669359927, 2.82519728467552799895120944827, 3.21636513982893502262985115531, 3.56594789470210554364709677015, 4.02646727411456748207857783367, 4.03128747738821136445894920105, 4.31908122275157766177104608443, 4.58911067733755502382119721115, 4.77539691794676714674716141375, 5.13266709729022515351766736213, 5.30729381166624692909130761414, 5.36293973070893895827501179779, 5.69811465275380996041629129026, 6.16780896932967141824504476078, 6.22700092173032153610153034138, 6.23820042770512513241332509429, 6.42361391667562408307510778561, 6.80034222082921771865042816481
