| L(s) = 1 | − 3·5-s − 13·7-s + 3·11-s + 30·17-s − 18·19-s − 36·23-s − 19·25-s + 102·29-s + 21·31-s + 39·35-s − 22·37-s − 20·43-s + 156·47-s + 120·49-s + 51·53-s − 9·55-s + 129·59-s + 120·61-s + 68·67-s − 36·73-s − 39·77-s + 125·79-s − 90·85-s − 126·89-s + 54·95-s − 36·101-s + 264·103-s + ⋯ |
| L(s) = 1 | − 3/5·5-s − 1.85·7-s + 3/11·11-s + 1.76·17-s − 0.947·19-s − 1.56·23-s − 0.759·25-s + 3.51·29-s + 0.677·31-s + 1.11·35-s − 0.594·37-s − 0.465·43-s + 3.31·47-s + 2.44·49-s + 0.962·53-s − 0.163·55-s + 2.18·59-s + 1.96·61-s + 1.01·67-s − 0.493·73-s − 0.506·77-s + 1.58·79-s − 1.05·85-s − 1.41·89-s + 0.568·95-s − 0.356·101-s + 2.56·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.103622804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.103622804\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + 13 T + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 28 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 112 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 30 T + 589 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T + 469 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 36 T + 767 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 51 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 21 T + 1108 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T - 885 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2774 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 156 T + 10321 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 51 T - 208 T^{2} - 51 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 129 T + 9028 T^{2} - 129 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 120 T + 8521 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 68 T + 135 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 36 T + 5761 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 125 T + 9384 T^{2} - 125 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9985 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 126 T + 13213 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \) |
| 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
−10.00103504462185217211871885139, −9.946909601642207278599281580131, −8.991837925948446470146329592337, −8.812290517819037095088383925921, −8.170565679185719463504945084380, −8.148883481733472859559548774470, −7.33134221068711295794784398652, −7.05467616036847374370608125162, −6.46924177594876001938000431096, −6.33409178180146633495235178842, −5.59870475382461759757586445451, −5.53445320764497538475048355806, −4.47970621571070185427691082913, −4.18591380239583960115763309400, −3.55245863448066359629770158001, −3.43349690593487288103811811113, −2.48952726479016198536709887010, −2.31741042390260209318408382009, −0.803481723064008126011251503253, −0.67926865930416439353280122756,
0.67926865930416439353280122756, 0.803481723064008126011251503253, 2.31741042390260209318408382009, 2.48952726479016198536709887010, 3.43349690593487288103811811113, 3.55245863448066359629770158001, 4.18591380239583960115763309400, 4.47970621571070185427691082913, 5.53445320764497538475048355806, 5.59870475382461759757586445451, 6.33409178180146633495235178842, 6.46924177594876001938000431096, 7.05467616036847374370608125162, 7.33134221068711295794784398652, 8.148883481733472859559548774470, 8.170565679185719463504945084380, 8.812290517819037095088383925921, 8.991837925948446470146329592337, 9.946909601642207278599281580131, 10.00103504462185217211871885139
