| L(s) = 1 | − 2·2-s + 4-s + 2·8-s − 8·11-s − 4·16-s + 16·22-s − 16·23-s + 8·25-s + 8·29-s + 2·32-s − 8·37-s − 16·43-s − 8·44-s + 32·46-s − 16·50-s − 8·53-s − 16·58-s + 3·64-s + 24·67-s + 16·74-s + 32·79-s + 32·86-s − 16·88-s − 16·92-s + 8·100-s + 16·106-s + 8·107-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.707·8-s − 2.41·11-s − 16-s + 3.41·22-s − 3.33·23-s + 8/5·25-s + 1.48·29-s + 0.353·32-s − 1.31·37-s − 2.43·43-s − 1.20·44-s + 4.71·46-s − 2.26·50-s − 1.09·53-s − 2.10·58-s + 3/8·64-s + 2.93·67-s + 1.85·74-s + 3.60·79-s + 3.45·86-s − 1.70·88-s − 1.66·92-s + 4/5·100-s + 1.55·106-s + 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1029839502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1029839502\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 16 T^{2} - 33 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 6 T^{2} - 325 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 62 T^{2} + 1635 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $C_2^3$ | \( 1 - 120 T^{2} + 10679 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 96 T^{2} + 3887 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 144 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.35020863070969568433319871394, −7.02470379203412692865593266867, −6.97637096296907818112749860725, −6.58300978484820659563953965345, −6.38536035546996578316951972389, −6.38102920478996826162573030590, −5.99002442487169939308747345805, −5.43850829454985606710629406143, −5.36162345824979576389825806372, −5.33030396604507299284561070714, −4.89475916383394509764909493805, −4.84937992039582270378873154830, −4.46517422454321559263041706085, −4.20300042044889463790303967070, −3.91256610908910115808884881165, −3.43221590736018038087150992813, −3.33757561573330531664897839130, −3.07946469170882912184006276212, −2.61626294299315091425606209258, −2.18695991845347059750656999089, −2.09738645884595459786518361971, −1.88932341775610905710275502031, −1.21843116791265542729606559891, −0.76092945166505975235038241402, −0.13634424550522366757976472100,
0.13634424550522366757976472100, 0.76092945166505975235038241402, 1.21843116791265542729606559891, 1.88932341775610905710275502031, 2.09738645884595459786518361971, 2.18695991845347059750656999089, 2.61626294299315091425606209258, 3.07946469170882912184006276212, 3.33757561573330531664897839130, 3.43221590736018038087150992813, 3.91256610908910115808884881165, 4.20300042044889463790303967070, 4.46517422454321559263041706085, 4.84937992039582270378873154830, 4.89475916383394509764909493805, 5.33030396604507299284561070714, 5.36162345824979576389825806372, 5.43850829454985606710629406143, 5.99002442487169939308747345805, 6.38102920478996826162573030590, 6.38536035546996578316951972389, 6.58300978484820659563953965345, 6.97637096296907818112749860725, 7.02470379203412692865593266867, 7.35020863070969568433319871394
