| L(s) = 1 | − 2·3-s + 4·7-s + 3·9-s + 8·19-s − 8·21-s − 6·25-s − 4·27-s + 4·29-s − 8·31-s − 4·37-s − 16·47-s + 9·49-s + 20·53-s − 16·57-s + 8·59-s + 12·63-s + 12·75-s + 5·81-s − 24·83-s − 8·87-s + 16·93-s + 24·103-s + 28·109-s + 8·111-s + 4·113-s − 6·121-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 9-s + 1.83·19-s − 1.74·21-s − 6/5·25-s − 0.769·27-s + 0.742·29-s − 1.43·31-s − 0.657·37-s − 2.33·47-s + 9/7·49-s + 2.74·53-s − 2.11·57-s + 1.04·59-s + 1.51·63-s + 1.38·75-s + 5/9·81-s − 2.63·83-s − 0.857·87-s + 1.65·93-s + 2.36·103-s + 2.68·109-s + 0.759·111-s + 0.376·113-s − 0.545·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.515117339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.515117339\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.391046533691483698004382219288, −8.222151966740788314456121575198, −7.47837202269369036994254596914, −7.17212008872630953090710242182, −6.96458755801090666740322092908, −6.03898188878233827131141159382, −5.56983356275681900785379754785, −5.47373462538208654239189473772, −4.77602747706575131165960838209, −4.51006367899790870186381301422, −3.73528919773956898681806633984, −3.22568568252812840477382664374, −2.09803110754033007053716721218, −1.61711448111568648295505994452, −0.75476969164145883416664038839,
0.75476969164145883416664038839, 1.61711448111568648295505994452, 2.09803110754033007053716721218, 3.22568568252812840477382664374, 3.73528919773956898681806633984, 4.51006367899790870186381301422, 4.77602747706575131165960838209, 5.47373462538208654239189473772, 5.56983356275681900785379754785, 6.03898188878233827131141159382, 6.96458755801090666740322092908, 7.17212008872630953090710242182, 7.47837202269369036994254596914, 8.222151966740788314456121575198, 8.391046533691483698004382219288
