| L(s) = 1 | + 4·7-s + 24·13-s + 64·31-s + 120·37-s + 216·43-s + 8·49-s + 216·61-s + 136·67-s + 260·73-s + 96·91-s + 276·97-s + 380·103-s + 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 288·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4/7·7-s + 1.84·13-s + 2.06·31-s + 3.24·37-s + 5.02·43-s + 8/49·49-s + 3.54·61-s + 2.02·67-s + 3.56·73-s + 1.05·91-s + 2.84·97-s + 3.68·103-s + 0.132·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.70·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(12.40194582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.40194582\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 160958 T^{4} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 526 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 248318 T^{4} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 1432 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 60 T + 1800 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 2362 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 108 T + 5832 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 9752638 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 10107038 T^{4} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 712 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 68 T + 2312 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 6082 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T + 8450 T^{2} - 130 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 818 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 8835358 T^{4} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 138 T + 9522 T^{2} - 138 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.13967431374169999519051363887, −6.54726669437684143149979597277, −6.50539773373611074330688371806, −6.39560056743709146448406725563, −6.27448797103342918570628434134, −5.92141130594485499831262711357, −5.52060556928902702599626362417, −5.46941069031287127105287705400, −5.46794946204761594117755254225, −4.79695281462318131155442300698, −4.58764731794192803271179355086, −4.43171332026839469679158168506, −4.24820761078319765218381005113, −3.98928067947143976707087028051, −3.52633155187517743274469051161, −3.48489463149326506970023839547, −3.29696460856281093164272371008, −2.51801254698134409589825625076, −2.33290479195413455745788471762, −2.31454198946793394899159778947, −2.19119707128953804563594307166, −1.13607913078108107570221487113, −1.02391828075681367834250441193, −0.801389046748807590973941495675, −0.71262810418280217787047583406,
0.71262810418280217787047583406, 0.801389046748807590973941495675, 1.02391828075681367834250441193, 1.13607913078108107570221487113, 2.19119707128953804563594307166, 2.31454198946793394899159778947, 2.33290479195413455745788471762, 2.51801254698134409589825625076, 3.29696460856281093164272371008, 3.48489463149326506970023839547, 3.52633155187517743274469051161, 3.98928067947143976707087028051, 4.24820761078319765218381005113, 4.43171332026839469679158168506, 4.58764731794192803271179355086, 4.79695281462318131155442300698, 5.46794946204761594117755254225, 5.46941069031287127105287705400, 5.52060556928902702599626362417, 5.92141130594485499831262711357, 6.27448797103342918570628434134, 6.39560056743709146448406725563, 6.50539773373611074330688371806, 6.54726669437684143149979597277, 7.13967431374169999519051363887
