| L(s) = 1 | − 4·2-s − 6·3-s + 4-s + 10·5-s + 24·6-s + 24·8-s + 27·9-s − 40·10-s − 92·11-s − 6·12-s − 8·13-s − 60·15-s − 47·16-s + 44·17-s − 108·18-s + 108·19-s + 10·20-s + 368·22-s − 320·23-s − 144·24-s + 75·25-s + 32·26-s − 108·27-s − 236·29-s + 240·30-s + 60·31-s + 52·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/8·4-s + 0.894·5-s + 1.63·6-s + 1.06·8-s + 9-s − 1.26·10-s − 2.52·11-s − 0.144·12-s − 0.170·13-s − 1.03·15-s − 0.734·16-s + 0.627·17-s − 1.41·18-s + 1.30·19-s + 0.111·20-s + 3.56·22-s − 2.90·23-s − 1.22·24-s + 3/5·25-s + 0.241·26-s − 0.769·27-s − 1.51·29-s + 1.46·30-s + 0.347·31-s + 0.287·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5167924456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5167924456\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p^{2} T + 15 T^{2} + p^{5} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 92 T + 4758 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T - 2810 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 44 T + 630 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 108 T + 13254 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 320 T + 47934 T^{2} + 320 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 236 T + 61982 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 60 T + 8462 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 204 T + 108830 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 44 T + 106326 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 136 T + 81718 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 400 T + 106526 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 260838 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 464 T + 458102 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 684 T + 535646 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 736 T + 602470 T^{2} - 736 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 740 T + 757502 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 424 T + 748558 T^{2} + 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 408 T - 143586 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 608 T + 1200710 T^{2} + 608 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1332 T + 1790774 T^{2} - 1332 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2448 T + 3286542 T^{2} - 2448 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−10.09822262855160232908277404681, −10.01149648381207986324179482669, −9.532379861310031990354168600605, −9.101716728312629157979502679994, −8.393794864784479190534610305826, −8.027410583642715039046823903395, −7.72065265714192429750247547816, −7.46432206005037281795695406249, −6.80818752292857090370165280033, −5.98002292838930365049363290212, −5.72722030850673878027078542275, −5.52330467876259091114755036029, −4.88937369512422208330708617597, −4.52672211296869160804231260173, −3.70316437819187515230033112303, −2.96786120939360784488264558663, −2.10565151223534747762414395514, −1.79845890558410588897509505609, −0.57884463335329817099894324883, −0.51949000523997107205478924213,
0.51949000523997107205478924213, 0.57884463335329817099894324883, 1.79845890558410588897509505609, 2.10565151223534747762414395514, 2.96786120939360784488264558663, 3.70316437819187515230033112303, 4.52672211296869160804231260173, 4.88937369512422208330708617597, 5.52330467876259091114755036029, 5.72722030850673878027078542275, 5.98002292838930365049363290212, 6.80818752292857090370165280033, 7.46432206005037281795695406249, 7.72065265714192429750247547816, 8.027410583642715039046823903395, 8.393794864784479190534610305826, 9.101716728312629157979502679994, 9.532379861310031990354168600605, 10.01149648381207986324179482669, 10.09822262855160232908277404681
