| L(s) = 1 | − 4-s + 2·11-s − 3·16-s + 4·19-s + 12·29-s + 24·31-s + 6·41-s − 2·44-s + 7·49-s − 48·59-s − 2·61-s + 3·64-s + 50·71-s − 4·76-s + 6·79-s − 18·89-s + 8·101-s + 8·109-s − 12·116-s − 33·121-s − 24·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.603·11-s − 3/4·16-s + 0.917·19-s + 2.22·29-s + 4.31·31-s + 0.937·41-s − 0.301·44-s + 49-s − 6.24·59-s − 0.256·61-s + 3/8·64-s + 5.93·71-s − 0.458·76-s + 0.675·79-s − 1.90·89-s + 0.796·101-s + 0.766·109-s − 1.11·116-s − 3·121-s − 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.243526103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.243526103\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T^{2} + p^{2} T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - p T^{2} + 4 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 59 T^{2} + 1444 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 43 T^{2} + 1176 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 31 T^{2} - 120 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 3 T + 80 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 136 T^{2} + 8254 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1870 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 131 T^{2} + 9564 T^{4} - 131 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 8766 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 25 T + 294 T^{2} - 25 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 3 T + 122 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 9 T + 160 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 41 T^{2} + 14032 T^{4} + 41 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.21114224045757105056633516331, −6.18025790019049081302577031680, −5.83134494950060801168964073531, −5.56897309007758295193014550877, −5.41345288048321806826107657821, −4.96770664861745775233450572262, −4.92981475752757559947787239089, −4.70138885877770536539598188081, −4.68054411481477592986656529344, −4.38556998991923175014353566161, −4.22486912221671974006574538213, −3.95595790217648379341153352371, −3.71095809048708001844004660012, −3.48500939126526816548951121997, −3.12827078831167771823960325303, −2.89237118834506148333026905213, −2.69569270250612422945595442605, −2.57499287530024891199950292993, −2.50936491346950758331541196951, −1.74959036685978740967320433124, −1.66074262494975236278290587920, −1.38175453805479972862930024276, −0.874020332012693529786356045394, −0.66462675180913818179018144230, −0.54342970061074038321036766984,
0.54342970061074038321036766984, 0.66462675180913818179018144230, 0.874020332012693529786356045394, 1.38175453805479972862930024276, 1.66074262494975236278290587920, 1.74959036685978740967320433124, 2.50936491346950758331541196951, 2.57499287530024891199950292993, 2.69569270250612422945595442605, 2.89237118834506148333026905213, 3.12827078831167771823960325303, 3.48500939126526816548951121997, 3.71095809048708001844004660012, 3.95595790217648379341153352371, 4.22486912221671974006574538213, 4.38556998991923175014353566161, 4.68054411481477592986656529344, 4.70138885877770536539598188081, 4.92981475752757559947787239089, 4.96770664861745775233450572262, 5.41345288048321806826107657821, 5.56897309007758295193014550877, 5.83134494950060801168964073531, 6.18025790019049081302577031680, 6.21114224045757105056633516331
