| L(s) = 1 | − 2·25-s − 40·31-s − 20·49-s + 40·79-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-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 + ⋯ |
| L(s) = 1 | − 2/5·25-s − 7.18·31-s − 2.85·49-s + 4.50·79-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 + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.106099075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106099075\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
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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
−6.91722019354896981780305752541, −6.43588585262053776106561951387, −6.41583314382976815217056542379, −6.20802209861040798475380504279, −5.87174692736383901084906681931, −5.72972707891141567591415584032, −5.47293951695257001550821668354, −5.10439557511298434918095008138, −5.05922089051886962017136704070, −5.05650150553598078445832468228, −4.72790816424941823974454912830, −4.08812953444809035284952661395, −4.07627506567451177434918870363, −3.72537846358905626740712313125, −3.70013735610571574852967408864, −3.32668795441333367350410428061, −3.28943928624846235276789453245, −2.85154044452311813440062938172, −2.41776521036936597091717866246, −2.06472893292625588889635423436, −1.89065068412923881265996587624, −1.62128297620107574046629900470, −1.52940768476845855193513294193, −0.63047446408072178130645993478, −0.26306110701649476689209283389,
0.26306110701649476689209283389, 0.63047446408072178130645993478, 1.52940768476845855193513294193, 1.62128297620107574046629900470, 1.89065068412923881265996587624, 2.06472893292625588889635423436, 2.41776521036936597091717866246, 2.85154044452311813440062938172, 3.28943928624846235276789453245, 3.32668795441333367350410428061, 3.70013735610571574852967408864, 3.72537846358905626740712313125, 4.07627506567451177434918870363, 4.08812953444809035284952661395, 4.72790816424941823974454912830, 5.05650150553598078445832468228, 5.05922089051886962017136704070, 5.10439557511298434918095008138, 5.47293951695257001550821668354, 5.72972707891141567591415584032, 5.87174692736383901084906681931, 6.20802209861040798475380504279, 6.41583314382976815217056542379, 6.43588585262053776106561951387, 6.91722019354896981780305752541
