| L(s) = 1 | − 4·4-s + 12·9-s − 4·11-s + 4·16-s + 36·19-s + 10·25-s − 48·36-s + 16·44-s − 12·49-s + 16·64-s − 144·76-s + 54·81-s − 48·99-s − 40·100-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 48·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 432·171-s + ⋯ |
| L(s) = 1 | − 2·4-s + 4·9-s − 1.20·11-s + 16-s + 8.25·19-s + 2·25-s − 8·36-s + 2.41·44-s − 1.71·49-s + 2·64-s − 16.5·76-s + 6·81-s − 4.82·99-s − 4·100-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 33.0·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.541435658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.541435658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - p T + p T^{2} )^{4}( 1 + p T + p T^{2} )^{4} \) |
| 11 | \( ( 1 + 2 T + p T^{2} )^{4}( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + 6 T^{2} - 133 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2}( 1 + 26 T^{2} + 147 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 29 | \( ( 1 - p T^{2} )^{8} \) |
| 31 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2}( 1 + 54 T^{2} + 1547 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 41 | \( ( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | \( ( 1 - p T^{2} )^{8} \) |
| 47 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2}( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 53 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2}( 1 - 74 T^{2} + 2667 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 59 | \( ( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - p T^{2} )^{8} \) |
| 73 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + p T^{2} )^{8} \) |
| 89 | \( ( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | \( ( 1 + p T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.21034759731477675395818614737, −5.16072785843438027594897673162, −5.13934382679463075715741599552, −4.91768792184638798279449746822, −4.84562906962931722277601769456, −4.83104493061610764680662383195, −4.31186020691732548076200522895, −4.22690419489868561526161562490, −4.05195120731653746869987009993, −4.00358022914855215524693602791, −3.92608484995460559842599185049, −3.70916961231326489597934472969, −3.49722136184200058934076204008, −3.05716308425456486106804095217, −3.05570335452654621975645627210, −3.02913769597745818547456351169, −2.87740903830420743672945003882, −2.66490093073276848254757827511, −2.28377147812032970817921842221, −1.74241139406730770926300403374, −1.56410236858035265913207586343, −1.33166250613210432152871658895, −1.25241925759984226402549138537, −0.967907417294934314647723046511, −0.70102546280214667658795974500,
0.70102546280214667658795974500, 0.967907417294934314647723046511, 1.25241925759984226402549138537, 1.33166250613210432152871658895, 1.56410236858035265913207586343, 1.74241139406730770926300403374, 2.28377147812032970817921842221, 2.66490093073276848254757827511, 2.87740903830420743672945003882, 3.02913769597745818547456351169, 3.05570335452654621975645627210, 3.05716308425456486106804095217, 3.49722136184200058934076204008, 3.70916961231326489597934472969, 3.92608484995460559842599185049, 4.00358022914855215524693602791, 4.05195120731653746869987009993, 4.22690419489868561526161562490, 4.31186020691732548076200522895, 4.83104493061610764680662383195, 4.84562906962931722277601769456, 4.91768792184638798279449746822, 5.13934382679463075715741599552, 5.16072785843438027594897673162, 5.21034759731477675395818614737
Plot not available for L-functions of degree greater than 10.
