L(s) = 1 | + 2·3-s + 3·9-s + 12·17-s − 8·23-s − 6·25-s + 4·27-s − 12·29-s + 8·43-s − 2·49-s + 24·51-s + 12·53-s − 12·61-s − 16·69-s − 12·75-s + 5·81-s − 24·87-s + 28·101-s − 32·103-s + 16·107-s − 20·113-s + 18·121-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 4·147-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s + 2.91·17-s − 1.66·23-s − 6/5·25-s + 0.769·27-s − 2.22·29-s + 1.21·43-s − 2/7·49-s + 3.36·51-s + 1.64·53-s − 1.53·61-s − 1.92·69-s − 1.38·75-s + 5/9·81-s − 2.57·87-s + 2.78·101-s − 3.15·103-s + 1.54·107-s − 1.88·113-s + 1.63·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.329·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16451136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16451136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.115473333\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.115473333\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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
−8.598521452864360955829862839767, −8.062544245983616492325864985018, −7.80135093654205926373464353434, −7.71717408960355757698576197183, −7.39745331628988825392471511402, −7.02595923762181946990791318183, −6.30424164455667052840375526384, −6.00349448940733312872871203508, −5.54753876115079261667719401602, −5.50894190077442623009113207277, −4.88763169369792174593110492005, −4.04329914482921857822801261330, −4.04211585991692898604170277347, −3.70695839801149713489726659863, −3.06956472516852432533359385506, −2.94033016352601970521626431241, −2.02473872664100069897206355721, −1.92788264923453754657444941782, −1.30503241743468739708169986950, −0.53877028193135650714978210832,
0.53877028193135650714978210832, 1.30503241743468739708169986950, 1.92788264923453754657444941782, 2.02473872664100069897206355721, 2.94033016352601970521626431241, 3.06956472516852432533359385506, 3.70695839801149713489726659863, 4.04211585991692898604170277347, 4.04329914482921857822801261330, 4.88763169369792174593110492005, 5.50894190077442623009113207277, 5.54753876115079261667719401602, 6.00349448940733312872871203508, 6.30424164455667052840375526384, 7.02595923762181946990791318183, 7.39745331628988825392471511402, 7.71717408960355757698576197183, 7.80135093654205926373464353434, 8.062544245983616492325864985018, 8.598521452864360955829862839767