| L(s) = 1 | − 18·9-s + 520·17-s + 464·25-s − 88·41-s − 1.36e3·49-s + 928·73-s + 243·81-s + 2.36e3·89-s + 3.52e3·97-s − 5.99e3·113-s + 1.79e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 9.36e3·153-s + 157-s + 163-s + 167-s − 428·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 7.41·17-s + 3.71·25-s − 0.335·41-s − 3.98·49-s + 1.48·73-s + 1/3·81-s + 2.81·89-s + 3.69·97-s − 4.98·113-s + 1.34·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 4.94·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.194·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.385863944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.385863944\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 232 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 684 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 898 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 214 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 130 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 6662 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 9466 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 5560 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 35820 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 99506 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 22 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 144614 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 94358 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 293704 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 213622 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 280510 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 471926 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 634214 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 232 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 753516 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 878510 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 590 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 882 T + p^{3} T^{2} )^{4} \) |
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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
−6.37483857583903621483519414463, −6.11245811627820793148370154787, −6.03602948135561880716216032177, −5.52289032812634570842986095480, −5.43768341353080900737545985581, −5.26580877239353923257178460604, −5.05840302839084840631330229718, −4.96128245297631595082869169084, −4.90888573382486964983862899368, −4.41644898222817029004484846234, −4.01414080647556123084315098528, −3.70165926820421038495035177022, −3.34963946128575852829773391284, −3.30659273025519275321536819118, −3.26194867063116463217275559796, −3.03485193744276896686832695019, −2.95246513544080706464294351950, −2.35059118208543711165545394825, −2.11171610069943052011589727307, −1.58395083206004208407376483352, −1.25410279748361155897891919316, −1.10461460457556109960468495402, −1.01725672757802072959901131305, −0.75411850734899950683706515542, −0.28399442506119167375049669044,
0.28399442506119167375049669044, 0.75411850734899950683706515542, 1.01725672757802072959901131305, 1.10461460457556109960468495402, 1.25410279748361155897891919316, 1.58395083206004208407376483352, 2.11171610069943052011589727307, 2.35059118208543711165545394825, 2.95246513544080706464294351950, 3.03485193744276896686832695019, 3.26194867063116463217275559796, 3.30659273025519275321536819118, 3.34963946128575852829773391284, 3.70165926820421038495035177022, 4.01414080647556123084315098528, 4.41644898222817029004484846234, 4.90888573382486964983862899368, 4.96128245297631595082869169084, 5.05840302839084840631330229718, 5.26580877239353923257178460604, 5.43768341353080900737545985581, 5.52289032812634570842986095480, 6.03602948135561880716216032177, 6.11245811627820793148370154787, 6.37483857583903621483519414463
