| L(s) = 1 | + 4·7-s + 4·9-s − 8·17-s + 16·23-s + 12·25-s + 8·41-s − 32·47-s + 10·49-s + 16·63-s + 32·71-s + 24·73-s − 32·79-s + 2·81-s − 8·89-s − 40·97-s + 32·103-s + 24·113-s − 32·119-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·153-s + 157-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 4/3·9-s − 1.94·17-s + 3.33·23-s + 12/5·25-s + 1.24·41-s − 4.66·47-s + 10/7·49-s + 2.01·63-s + 3.79·71-s + 2.80·73-s − 3.60·79-s + 2/9·81-s − 0.847·89-s − 4.06·97-s + 3.15·103-s + 2.25·113-s − 2.93·119-s + 1.09·121-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 − 2.58·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.530994227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.530994227\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2^2:C_4$ | \( 1 - 4 T^{2} + 14 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 12 T^{2} + 78 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 - 12 T^{2} + 150 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 12 T^{2} - 18 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_4$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 36 T^{2} + 1038 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 44 T^{2} + 2038 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 116 T^{2} + 5974 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2:C_4$ | \( 1 - 140 T^{2} + 8470 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 84 T^{2} + 5334 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 196 T^{2} + 16174 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 44 T^{2} + 5614 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 204 T^{2} + 18870 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 132 T^{2} + 13134 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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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
−7.00836544098240935592014370262, −6.98217485171851867756057303295, −6.96465473286291381651267397979, −6.69790138617319049450406610339, −6.55201049442649774859874188828, −6.25135149903560284433546345555, −5.82313554600301867329904951194, −5.54211991132827173148920303965, −5.25256743639434861428355054113, −4.97410486289329159355585511665, −4.96338075279898886706201487179, −4.60944708084158395055226544674, −4.57121584337150750018229324129, −4.33101888893388265923544817982, −4.07102037604093919479035763842, −3.49319425413048393534753167581, −3.34162436164300955239520925699, −3.00978941287447457524284359054, −2.85505933486014992029168659074, −2.29153649005241271250802659531, −2.14023594186934743079211959290, −1.65434548895765472842795280294, −1.42026803324858544242387101314, −0.996958915264765278336784276167, −0.64624383288167329175721718935,
0.64624383288167329175721718935, 0.996958915264765278336784276167, 1.42026803324858544242387101314, 1.65434548895765472842795280294, 2.14023594186934743079211959290, 2.29153649005241271250802659531, 2.85505933486014992029168659074, 3.00978941287447457524284359054, 3.34162436164300955239520925699, 3.49319425413048393534753167581, 4.07102037604093919479035763842, 4.33101888893388265923544817982, 4.57121584337150750018229324129, 4.60944708084158395055226544674, 4.96338075279898886706201487179, 4.97410486289329159355585511665, 5.25256743639434861428355054113, 5.54211991132827173148920303965, 5.82313554600301867329904951194, 6.25135149903560284433546345555, 6.55201049442649774859874188828, 6.69790138617319049450406610339, 6.96465473286291381651267397979, 6.98217485171851867756057303295, 7.00836544098240935592014370262
