| L(s) = 1 | + 4·4-s − 6·5-s + 18·9-s + 10·13-s + 90·17-s − 24·20-s + 25·25-s + 42·29-s + 72·36-s − 210·37-s − 18·41-s − 108·45-s + 98·49-s + 40·52-s + 22·61-s − 64·64-s − 60·65-s + 360·68-s + 81·81-s − 540·85-s + 156·89-s + 100·100-s − 198·101-s + 728·109-s − 90·113-s + 168·116-s + 180·117-s + ⋯ |
| L(s) = 1 | + 4-s − 6/5·5-s + 2·9-s + 0.769·13-s + 5.29·17-s − 6/5·20-s + 25-s + 1.44·29-s + 2·36-s − 5.67·37-s − 0.439·41-s − 2.39·45-s + 2·49-s + 0.769·52-s + 0.360·61-s − 64-s − 0.923·65-s + 5.29·68-s + 81-s − 6.35·85-s + 1.75·89-s + 100-s − 1.96·101-s + 6.67·109-s − 0.796·113-s + 1.44·116-s + 1.53·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.687565085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.687565085\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 11 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 10 T - 69 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2}( 1 - 30 T + 611 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 19 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{2}( 1 + 42 T + 923 T^{2} + 42 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2}( 1 + 70 T + 3531 T^{2} + 70 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2}( 1 - 18 T - 1357 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 90 T + 5291 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} )( 1 + 90 T + 5291 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T - 3237 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 67 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 110 T + 6771 T^{2} - 110 p^{2} T^{3} + p^{4} T^{4} )( 1 + 110 T + 6771 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 78 T - 1837 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 130 T + 7491 T^{2} - 130 p^{2} T^{3} + p^{4} T^{4} )( 1 + 130 T + 7491 T^{2} + 130 p^{2} T^{3} + p^{4} T^{4} ) \) |
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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
−8.554103262142764066487476862208, −8.009446399783294623683094226895, −7.918064617605412561088511778324, −7.69450908800977527673549143222, −7.34474133065604530604538331195, −7.20855964465770187150150860468, −7.16936137710157066623208466866, −6.77478385700480393239100334499, −6.38223586144987132659605770716, −6.29529396352615069931334956778, −5.61851658764977896489860502410, −5.53533044307765026384517624621, −5.27036245955811571135924427938, −4.85121618525060407372046438200, −4.80500181494865447767346400627, −3.96541014856944324565641089327, −3.89015651130585456334065639553, −3.56537111052886649744038977824, −3.35116257332572857604456074203, −3.12922728794288230128280971845, −2.60153537523471249691806901933, −1.77981857795754577936028157743, −1.49192940524392675902003496721, −1.21187432375647220470528823228, −0.67623023498913613680095201143,
0.67623023498913613680095201143, 1.21187432375647220470528823228, 1.49192940524392675902003496721, 1.77981857795754577936028157743, 2.60153537523471249691806901933, 3.12922728794288230128280971845, 3.35116257332572857604456074203, 3.56537111052886649744038977824, 3.89015651130585456334065639553, 3.96541014856944324565641089327, 4.80500181494865447767346400627, 4.85121618525060407372046438200, 5.27036245955811571135924427938, 5.53533044307765026384517624621, 5.61851658764977896489860502410, 6.29529396352615069931334956778, 6.38223586144987132659605770716, 6.77478385700480393239100334499, 7.16936137710157066623208466866, 7.20855964465770187150150860468, 7.34474133065604530604538331195, 7.69450908800977527673549143222, 7.918064617605412561088511778324, 8.009446399783294623683094226895, 8.554103262142764066487476862208
