| L(s) = 1 | − 6·3-s + 6·7-s + 18·9-s − 24·11-s − 24·13-s + 24·17-s − 36·21-s − 6·23-s − 54·27-s + 16·31-s + 144·33-s − 96·37-s + 144·39-s − 96·41-s + 54·43-s − 54·47-s + 18·49-s − 144·51-s − 24·53-s + 64·61-s + 108·63-s + 6·67-s + 36·69-s + 96·71-s − 24·73-s − 144·77-s + 243·81-s + ⋯ |
| L(s) = 1 | − 2·3-s + 6/7·7-s + 2·9-s − 2.18·11-s − 1.84·13-s + 1.41·17-s − 1.71·21-s − 0.260·23-s − 2·27-s + 0.516·31-s + 4.36·33-s − 2.59·37-s + 3.69·39-s − 2.34·41-s + 1.25·43-s − 1.14·47-s + 0.367·49-s − 2.82·51-s − 0.452·53-s + 1.04·61-s + 12/7·63-s + 6/67·67-s + 0.521·69-s + 1.35·71-s − 0.328·73-s − 1.87·77-s + 3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.005731264611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.005731264611\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 782 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 96 T + 4608 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 54 T + 1458 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 54 T + 1458 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3362 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10882 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 186 T + 17298 T^{2} + 186 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14942 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.62142622361110683964806927791, −10.66954490905987829071683939586, −10.53863784305595026601911289852, −10.07728979633254194466892040913, −9.890973262221863511903444818723, −9.142369465324382375503828472157, −8.162287671038947030996512546941, −8.010771623259069212493028474325, −7.60398465311913779668310134852, −6.85063052334381933426249970897, −6.73298509886092704353022166965, −5.53399983855826140076185399419, −5.52132821086858525552392891958, −5.05508906680154453918389401408, −4.96124048103725630652438136674, −3.99532403397664104925185186807, −3.05468756678898112400383089991, −2.30047464617709442875287968907, −1.43532873878352851012003742086, −0.03758768381088179741434813751,
0.03758768381088179741434813751, 1.43532873878352851012003742086, 2.30047464617709442875287968907, 3.05468756678898112400383089991, 3.99532403397664104925185186807, 4.96124048103725630652438136674, 5.05508906680154453918389401408, 5.52132821086858525552392891958, 5.53399983855826140076185399419, 6.73298509886092704353022166965, 6.85063052334381933426249970897, 7.60398465311913779668310134852, 8.010771623259069212493028474325, 8.162287671038947030996512546941, 9.142369465324382375503828472157, 9.890973262221863511903444818723, 10.07728979633254194466892040913, 10.53863784305595026601911289852, 10.66954490905987829071683939586, 11.62142622361110683964806927791
