| L(s) = 1 | − 6·13-s + 4·17-s + 8·23-s + 6·25-s + 20·29-s − 8·43-s + 10·49-s + 12·53-s + 4·61-s − 4·101-s + 32·103-s + 16·107-s − 28·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 23·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.66·13-s + 0.970·17-s + 1.66·23-s + 6/5·25-s + 3.71·29-s − 1.21·43-s + 10/7·49-s + 1.64·53-s + 0.512·61-s − 0.398·101-s + 3.15·103-s + 1.54·107-s − 2.63·113-s + 2·121-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 + 1.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.851407721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.851407721\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.361995907518608237787438983653, −9.007385919229748411351164434573, −8.553501971817179384632095504635, −8.447263688715366358683879441083, −7.77692294713283931596655897320, −7.48497016770577177653000297543, −6.93719216554941773200738777003, −6.78018313711248701755063961250, −6.45375414631718386441229477036, −5.70819126897290076721028271593, −5.22682445916237200712828707122, −5.06156645094311968166905732233, −4.48232571967047506210393698411, −4.33172448019589934184046759446, −3.21863537722915619675050225897, −3.16537262543540868017865575838, −2.59868470270459072400301490415, −2.13198175037760075101119464600, −1.02058498317847053628252469040, −0.794807746838400400493292765771,
0.794807746838400400493292765771, 1.02058498317847053628252469040, 2.13198175037760075101119464600, 2.59868470270459072400301490415, 3.16537262543540868017865575838, 3.21863537722915619675050225897, 4.33172448019589934184046759446, 4.48232571967047506210393698411, 5.06156645094311968166905732233, 5.22682445916237200712828707122, 5.70819126897290076721028271593, 6.45375414631718386441229477036, 6.78018313711248701755063961250, 6.93719216554941773200738777003, 7.48497016770577177653000297543, 7.77692294713283931596655897320, 8.447263688715366358683879441083, 8.553501971817179384632095504635, 9.007385919229748411351164434573, 9.361995907518608237787438983653
