| L(s) = 1 | − 20·25-s − 64·67-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 251-s + ⋯ |
| L(s) = 1 | − 4·25-s − 7.81·67-s − 4·121-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 + 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 + 0.0646·239-s + 0.0644·241-s + 0.0631·251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 146 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 194 T^{4} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 2062 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2^3$ | \( 1 - 1966 T^{4} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 7682 T^{4} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 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
−5.80364309809751450505556660585, −5.75083672563839655216111551405, −5.33346530256710295936851411287, −5.26227966166048988244772824239, −5.21114722170712189196749862302, −4.69973857411545586297705161694, −4.65148409012182754932447750642, −4.50558026123186279143574713940, −4.25160323299228911034355326772, −4.21742800480488082328991502184, −3.85631421440277869096500323868, −3.81538375692025753991748620165, −3.63510243770382464811985115218, −3.29487586134192794883138674384, −3.11958349431956666006985334246, −2.99891639345788159405111319893, −2.83198476199136641689326549927, −2.34404997446115660914470319164, −2.27196005744173555007763847150, −2.09787078974469372903157306254, −2.02828459907767621748096648731, −1.43425819682647374049095168760, −1.34925923964010256885618909609, −1.21377998738872618041691397987, −1.08862081458042372228834389055, 0, 0, 0, 0,
1.08862081458042372228834389055, 1.21377998738872618041691397987, 1.34925923964010256885618909609, 1.43425819682647374049095168760, 2.02828459907767621748096648731, 2.09787078974469372903157306254, 2.27196005744173555007763847150, 2.34404997446115660914470319164, 2.83198476199136641689326549927, 2.99891639345788159405111319893, 3.11958349431956666006985334246, 3.29487586134192794883138674384, 3.63510243770382464811985115218, 3.81538375692025753991748620165, 3.85631421440277869096500323868, 4.21742800480488082328991502184, 4.25160323299228911034355326772, 4.50558026123186279143574713940, 4.65148409012182754932447750642, 4.69973857411545586297705161694, 5.21114722170712189196749862302, 5.26227966166048988244772824239, 5.33346530256710295936851411287, 5.75083672563839655216111551405, 5.80364309809751450505556660585
