| L(s) = 1 | + 4·4-s − 8·13-s + 4·16-s + 8·25-s − 8·37-s − 28·49-s − 32·52-s − 20·61-s − 16·64-s + 64·73-s − 32·97-s + 32·100-s − 80·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2·4-s − 2.21·13-s + 16-s + 8/5·25-s − 1.31·37-s − 4·49-s − 4.43·52-s − 2.56·61-s − 2·64-s + 7.49·73-s − 3.24·97-s + 16/5·100-s − 7.66·109-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.816211629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.816211629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2}( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 7 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 4 T + p T^{2} )^{4}( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | \( ( 1 - 16 T^{2} - 33 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - p T^{2} )^{8} \) |
| 23 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2}( 1 - 40 T^{2} + 759 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 31 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 80 T^{2} + 4719 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 10 T + p T^{2} )^{4}( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + p T^{2} )^{8} \) |
| 73 | \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 160 T^{2} + 17679 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−5.10781939722747765319429414615, −5.04221340443101416748257459680, −5.03982505553591932495899511758, −4.88578337414475415311172885628, −4.66938129933162497760007815262, −4.48379095323540737101875717723, −4.25561315805366334868006020343, −4.16376931893620988673895029856, −3.97847739302993635595370724472, −3.83498176634655985044399833025, −3.52620164722905384156990913473, −3.49743207878092520574205188303, −3.08241456519540723097113642259, −2.95488808405698567506165412481, −2.88063652577383678518508707843, −2.84204931387701837945935239171, −2.69895940520614535449122510669, −2.41355534868017725997776568560, −2.04050369705806117574202086766, −1.94901573889275965182356239732, −1.79709734849856550387382991267, −1.69829120998601007108540791821, −1.34691418485711503987204381741, −0.883837178750420943295331817178, −0.28733921152702162984702211301,
0.28733921152702162984702211301, 0.883837178750420943295331817178, 1.34691418485711503987204381741, 1.69829120998601007108540791821, 1.79709734849856550387382991267, 1.94901573889275965182356239732, 2.04050369705806117574202086766, 2.41355534868017725997776568560, 2.69895940520614535449122510669, 2.84204931387701837945935239171, 2.88063652577383678518508707843, 2.95488808405698567506165412481, 3.08241456519540723097113642259, 3.49743207878092520574205188303, 3.52620164722905384156990913473, 3.83498176634655985044399833025, 3.97847739302993635595370724472, 4.16376931893620988673895029856, 4.25561315805366334868006020343, 4.48379095323540737101875717723, 4.66938129933162497760007815262, 4.88578337414475415311172885628, 5.03982505553591932495899511758, 5.04221340443101416748257459680, 5.10781939722747765319429414615
Plot not available for L-functions of degree greater than 10.
