| L(s) = 1 | − 12·7-s + 12·23-s − 2·25-s + 24·31-s − 24·41-s − 12·47-s + 68·49-s + 24·71-s − 16·73-s + 48·79-s + 24·89-s + 16·97-s − 12·103-s + 24·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 144·161-s + 163-s + 167-s + 28·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4.53·7-s + 2.50·23-s − 2/5·25-s + 4.31·31-s − 3.74·41-s − 1.75·47-s + 68/7·49-s + 2.84·71-s − 1.87·73-s + 5.40·79-s + 2.54·89-s + 1.62·97-s − 1.18·103-s + 2.25·113-s + 1.81·121-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 + 0.0798·157-s − 11.3·161-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.047918167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.047918167\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2154 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 13002 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.46312270284939233053743924455, −6.13353844082001422107185555589, −5.92956748791203245847954121089, −5.80909311151904962056639811651, −5.41762659699721921527286959763, −5.00945240270312655905294740940, −4.95009263289322932831623936628, −4.89300288987334518846189442302, −4.67567359779411195895236525783, −4.45805377874823238036654968758, −3.88471281545415381491634780591, −3.74248122743724503558144945626, −3.49815388047164014256668781848, −3.39720505041604725841037156195, −3.25663737102043192800435737279, −3.09328418056973568516646821151, −2.89838561759075302840972019915, −2.68061071083011225059821945717, −2.21837508841751435191391311767, −2.20654698750174407324397915462, −1.70508584190718889663602764046, −1.21742491526225121391326462421, −0.74454973611065485979237099178, −0.57066619521290812419596810377, −0.40610297949479450680656394433,
0.40610297949479450680656394433, 0.57066619521290812419596810377, 0.74454973611065485979237099178, 1.21742491526225121391326462421, 1.70508584190718889663602764046, 2.20654698750174407324397915462, 2.21837508841751435191391311767, 2.68061071083011225059821945717, 2.89838561759075302840972019915, 3.09328418056973568516646821151, 3.25663737102043192800435737279, 3.39720505041604725841037156195, 3.49815388047164014256668781848, 3.74248122743724503558144945626, 3.88471281545415381491634780591, 4.45805377874823238036654968758, 4.67567359779411195895236525783, 4.89300288987334518846189442302, 4.95009263289322932831623936628, 5.00945240270312655905294740940, 5.41762659699721921527286959763, 5.80909311151904962056639811651, 5.92956748791203245847954121089, 6.13353844082001422107185555589, 6.46312270284939233053743924455
