| L(s) = 1 | + 14·4-s + 4·9-s + 8·11-s + 115·16-s + 88·29-s + 56·36-s + 112·44-s − 98·49-s + 700·64-s + 248·71-s − 152·79-s − 150·81-s + 32·99-s + 568·109-s + 1.23e3·116-s − 444·121-s + 127-s + 131-s + 137-s + 139-s + 460·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 316·169-s + ⋯ |
| L(s) = 1 | + 7/2·4-s + 4/9·9-s + 8/11·11-s + 7.18·16-s + 3.03·29-s + 14/9·36-s + 2.54·44-s − 2·49-s + 10.9·64-s + 3.49·71-s − 1.92·79-s − 1.85·81-s + 0.323·99-s + 5.21·109-s + 10.6·116-s − 3.66·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.19·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.86·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(9.873777028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.873777028\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{4} T^{4} )^{2} \) |
| 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 542 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 382 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 958 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2542 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2642 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2542 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 3698 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 4462 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5342 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 1378 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 8782 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 7778 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 38 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 12158 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15122 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 18098 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−9.315060605194204751357081247292, −8.480733101903409544576083552726, −8.331276078610471621449477997621, −8.319700406401719198824630685979, −8.021195763622993339379366735371, −7.40386059078576089930030869824, −7.32834370949643430570017000516, −7.13043378388297136544258876790, −6.74770484874808900913817841216, −6.71022549898914785469855203783, −6.20054814170145077959704121991, −6.10124399061851031328314428532, −6.08600948646534096598894660767, −5.44352610278929527852537745025, −4.98182056837784220657325408682, −4.78189359579872758890509204845, −4.26063865504008912041474479826, −3.77267025881394423684502810258, −3.29121019107990596362902596636, −3.09657233747427090374506067957, −2.72588040412026452917495219098, −2.18245707412661714132154404108, −2.05522890169869190746273426585, −1.20380124664691249088974293876, −1.18156816150138094970852961179,
1.18156816150138094970852961179, 1.20380124664691249088974293876, 2.05522890169869190746273426585, 2.18245707412661714132154404108, 2.72588040412026452917495219098, 3.09657233747427090374506067957, 3.29121019107990596362902596636, 3.77267025881394423684502810258, 4.26063865504008912041474479826, 4.78189359579872758890509204845, 4.98182056837784220657325408682, 5.44352610278929527852537745025, 6.08600948646534096598894660767, 6.10124399061851031328314428532, 6.20054814170145077959704121991, 6.71022549898914785469855203783, 6.74770484874808900913817841216, 7.13043378388297136544258876790, 7.32834370949643430570017000516, 7.40386059078576089930030869824, 8.021195763622993339379366735371, 8.319700406401719198824630685979, 8.331276078610471621449477997621, 8.480733101903409544576083552726, 9.315060605194204751357081247292
