| L(s) = 1 | + 4-s + 9-s + 8·11-s + 2·19-s − 16·29-s + 36-s + 48·41-s + 8·44-s − 11·49-s + 12·59-s + 26·61-s − 64-s + 64·71-s + 2·76-s + 26·79-s − 4·89-s + 8·99-s − 24·101-s − 38·109-s − 16·116-s + 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s + 2.41·11-s + 0.458·19-s − 2.97·29-s + 1/6·36-s + 7.49·41-s + 1.20·44-s − 1.57·49-s + 1.56·59-s + 3.32·61-s − 1/8·64-s + 7.59·71-s + 0.229·76-s + 2.92·79-s − 0.423·89-s + 0.804·99-s − 2.38·101-s − 3.63·109-s − 1.48·116-s + 3.45·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.943253163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.943253163\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 102 T^{2} + 7595 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 25 T^{2} - 4704 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 95 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.15113659779806067395518645274, −6.69322319013603972494551943242, −6.67022872828193978723763717774, −6.55311697399366573876069263881, −6.28182236583458764167666509662, −5.94048818279821578484593436124, −5.88796281321219376921436782687, −5.43542562505746363398967364027, −5.29963376985188892629310228706, −5.22880299949709666143004636516, −4.85144107346652865181770554042, −4.37994642556282337517492780439, −4.12130299811634781680125576415, −3.93424177375578460057053015644, −3.81542618264262698184348260467, −3.63767580109923546009208567831, −3.59848743668428174027061162901, −2.79877002595630844501861641770, −2.53299316946435144913347081046, −2.35873320951065185349454212564, −2.06684590099168519849105751109, −1.84262058371971812591948548075, −1.09796300587227821767992319530, −0.934352102489432525779558567487, −0.75396134301424296751977966664,
0.75396134301424296751977966664, 0.934352102489432525779558567487, 1.09796300587227821767992319530, 1.84262058371971812591948548075, 2.06684590099168519849105751109, 2.35873320951065185349454212564, 2.53299316946435144913347081046, 2.79877002595630844501861641770, 3.59848743668428174027061162901, 3.63767580109923546009208567831, 3.81542618264262698184348260467, 3.93424177375578460057053015644, 4.12130299811634781680125576415, 4.37994642556282337517492780439, 4.85144107346652865181770554042, 5.22880299949709666143004636516, 5.29963376985188892629310228706, 5.43542562505746363398967364027, 5.88796281321219376921436782687, 5.94048818279821578484593436124, 6.28182236583458764167666509662, 6.55311697399366573876069263881, 6.67022872828193978723763717774, 6.69322319013603972494551943242, 7.15113659779806067395518645274
