| L(s) = 1 | + 6·11-s − 72·23-s + 47·25-s + 102·29-s + 44·37-s + 20·43-s − 102·53-s + 136·67-s + 250·79-s + 66·107-s − 64·109-s + 372·113-s − 215·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 290·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 6/11·11-s − 3.13·23-s + 1.87·25-s + 3.51·29-s + 1.18·37-s + 0.465·43-s − 1.92·53-s + 2.02·67-s + 3.16·79-s + 0.616·107-s − 0.587·109-s + 3.29·113-s − 1.77·121-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 + 1.71·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.653947122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.653947122\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 47 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 278 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 614 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 36 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 51 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1775 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2774 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3694 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 51 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 1415 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2642 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 68 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 10226 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 125 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 9985 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 10550 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2857 T^{2} + p^{4} T^{4} \) |
| 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
−9.294923364377489864069047184090, −8.906652089628221075596733083031, −8.392256781374825300670357944517, −8.161190045415239265226422677538, −7.904393579619229456098004208559, −7.41067751098426083061866196304, −6.65145948139757325923814391078, −6.51847016329701333011891492747, −6.29090405627853752433323194801, −5.81389419990385960317617490177, −5.07559277296050811818758600777, −4.83734395990131700492667741411, −4.20986169073650742352224124075, −4.12773637424421806406480532402, −3.22625238301243195602922731267, −3.00612723864448173693375679947, −2.25616239428369658144311135348, −1.88560752250331721179155328289, −0.923737939780226743608324518512, −0.64209959940876811606043829538,
0.64209959940876811606043829538, 0.923737939780226743608324518512, 1.88560752250331721179155328289, 2.25616239428369658144311135348, 3.00612723864448173693375679947, 3.22625238301243195602922731267, 4.12773637424421806406480532402, 4.20986169073650742352224124075, 4.83734395990131700492667741411, 5.07559277296050811818758600777, 5.81389419990385960317617490177, 6.29090405627853752433323194801, 6.51847016329701333011891492747, 6.65145948139757325923814391078, 7.41067751098426083061866196304, 7.904393579619229456098004208559, 8.161190045415239265226422677538, 8.392256781374825300670357944517, 8.906652089628221075596733083031, 9.294923364377489864069047184090
