| L(s) = 1 | − 3·4-s − 3·9-s + 4·11-s + 4·16-s + 32·19-s − 2·29-s + 9·36-s − 10·41-s − 12·44-s − 5·49-s − 28·59-s − 14·61-s − 9·64-s + 8·71-s − 96·76-s − 12·79-s + 60·89-s − 12·99-s + 36·101-s − 20·109-s + 6·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 9-s + 1.20·11-s + 16-s + 7.34·19-s − 0.371·29-s + 3/2·36-s − 1.56·41-s − 1.80·44-s − 5/7·49-s − 3.64·59-s − 1.79·61-s − 9/8·64-s + 0.949·71-s − 11.0·76-s − 1.35·79-s + 6.35·89-s − 1.20·99-s + 3.58·101-s − 1.91·109-s + 0.557·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.313395972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313395972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} ) \) |
| 47 | $C_2^3$ | \( 1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 85 T^{2} + 336 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.062248999002875606993402376261, −8.973711634705946568150305396738, −8.578638647989805930346787557363, −7.968780208217688930806018493554, −7.912352711906511156333769662445, −7.51386289938755394923643911101, −7.47395685017942830436648964286, −7.43938691140677565904046477327, −6.79013835701274226589800914758, −6.53159072992239292963102843242, −5.97373834221418650207816815852, −5.81671718120177359628184083453, −5.75790584858429441523721573098, −5.08743014639196802501939699801, −5.01176078281149394015673956521, −4.82443299653086056644195378616, −4.70770831648353199562520420336, −3.87077696061692530105290520323, −3.49208418543190910432014017075, −3.27877565443137169676674328242, −3.20299101371822093947110918609, −2.92472370356700460003154624998, −1.77190946552616613835539648406, −1.29092685980446188214799950289, −0.815525097298749108345657997160,
0.815525097298749108345657997160, 1.29092685980446188214799950289, 1.77190946552616613835539648406, 2.92472370356700460003154624998, 3.20299101371822093947110918609, 3.27877565443137169676674328242, 3.49208418543190910432014017075, 3.87077696061692530105290520323, 4.70770831648353199562520420336, 4.82443299653086056644195378616, 5.01176078281149394015673956521, 5.08743014639196802501939699801, 5.75790584858429441523721573098, 5.81671718120177359628184083453, 5.97373834221418650207816815852, 6.53159072992239292963102843242, 6.79013835701274226589800914758, 7.43938691140677565904046477327, 7.47395685017942830436648964286, 7.51386289938755394923643911101, 7.912352711906511156333769662445, 7.968780208217688930806018493554, 8.578638647989805930346787557363, 8.973711634705946568150305396738, 9.062248999002875606993402376261
