| L(s) = 1 | − 12·19-s − 8·31-s + 26·49-s − 4·61-s + 12·79-s − 64·109-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 2.75·19-s − 1.43·31-s + 26/7·49-s − 0.512·61-s + 1.35·79-s − 6.13·109-s − 2.18·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.61·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.135720732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.135720732\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 73 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 72 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 84 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 193 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.86528528412522724079353843836, −7.59311716056863857875950867392, −7.19536141654616463916172647649, −7.06336344421532421814038642601, −6.87961662612998868504155277168, −6.72201669085319595487951365473, −6.25582017012099008313409589401, −6.23697560124964681076913275600, −5.69349141421041870368883655237, −5.64314905237052058772885131642, −5.54945835612603859208996413079, −4.97329274697948220646736498621, −4.91354576492017427015098996358, −4.29503185762542678734933904786, −4.21871151075220772968317566886, −4.14142567463601568673380914962, −3.68838905701190899944937263707, −3.51083706802360544452006946189, −2.98148059706059539763732947131, −2.55891320612509104968977451475, −2.40614944817258005486158744738, −2.04062455265025563980423584219, −1.68221706706578788417056345830, −1.11141292196279116917135978453, −0.35332328426712925442445266471,
0.35332328426712925442445266471, 1.11141292196279116917135978453, 1.68221706706578788417056345830, 2.04062455265025563980423584219, 2.40614944817258005486158744738, 2.55891320612509104968977451475, 2.98148059706059539763732947131, 3.51083706802360544452006946189, 3.68838905701190899944937263707, 4.14142567463601568673380914962, 4.21871151075220772968317566886, 4.29503185762542678734933904786, 4.91354576492017427015098996358, 4.97329274697948220646736498621, 5.54945835612603859208996413079, 5.64314905237052058772885131642, 5.69349141421041870368883655237, 6.23697560124964681076913275600, 6.25582017012099008313409589401, 6.72201669085319595487951365473, 6.87961662612998868504155277168, 7.06336344421532421814038642601, 7.19536141654616463916172647649, 7.59311716056863857875950867392, 7.86528528412522724079353843836
