| L(s) = 1 | + 4·2-s + 4·4-s − 10·5-s + 12·7-s − 16·8-s − 18·9-s − 40·10-s − 18·11-s + 12·13-s + 48·14-s − 64·16-s − 72·18-s − 2·19-s − 40·20-s − 72·22-s − 26·23-s + 25·25-s + 48·26-s + 48·28-s − 64·32-s − 120·35-s − 72·36-s − 54·37-s − 8·38-s + 160·40-s + 156·41-s − 72·44-s + ⋯ |
| L(s) = 1 | + 2·2-s + 4-s − 2·5-s + 12/7·7-s − 2·8-s − 2·9-s − 4·10-s − 1.63·11-s + 0.923·13-s + 24/7·14-s − 4·16-s − 4·18-s − 0.105·19-s − 2·20-s − 3.27·22-s − 1.13·23-s + 25-s + 1.84·26-s + 12/7·28-s − 2·32-s − 3.42·35-s − 2·36-s − 1.45·37-s − 0.210·38-s + 4·40-s + 3.80·41-s − 1.63·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.154246280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.154246280\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2}( 1 - 18 T + 203 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T - 133 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2}( 1 - 2 T - 357 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2}( 1 - 26 T + 147 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 54 T + p^{2} T^{2} )^{2}( 1 - 54 T + 1547 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T + 4403 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 - 86 T + p^{2} T^{2} )^{2}( 1 + 86 T + 5187 T^{2} + 86 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2}( 1 + 74 T + 2667 T^{2} + 74 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 78 T + 2603 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 18 T - 7597 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 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
−8.198698155605665546169179274492, −8.052697561565362300356543946661, −8.049226932103273779527137714584, −7.60219208228645671880275015653, −7.49138483742766372941795586260, −7.33215615418292279930427556593, −6.57602759954576783915760175705, −6.41050008376229076382858997058, −6.01315039349705164180554494845, −5.72081587808780655131721750281, −5.61041732813193889212236941804, −5.51316484170197457974708968047, −5.06019442726844714527868593035, −4.79363642966907114932456948932, −4.29031864306355805755314284795, −4.27021809883763046253865432242, −4.22274678780060166640087942535, −3.55664903031652946349427202314, −3.40943274586551830011353624082, −3.09326130544361853742910590059, −2.54886778668514132000994234095, −2.48386478225584725770365719531, −1.84864570371584823682222833516, −0.73013912661473500945057188285, −0.26355280092863787216749643539,
0.26355280092863787216749643539, 0.73013912661473500945057188285, 1.84864570371584823682222833516, 2.48386478225584725770365719531, 2.54886778668514132000994234095, 3.09326130544361853742910590059, 3.40943274586551830011353624082, 3.55664903031652946349427202314, 4.22274678780060166640087942535, 4.27021809883763046253865432242, 4.29031864306355805755314284795, 4.79363642966907114932456948932, 5.06019442726844714527868593035, 5.51316484170197457974708968047, 5.61041732813193889212236941804, 5.72081587808780655131721750281, 6.01315039349705164180554494845, 6.41050008376229076382858997058, 6.57602759954576783915760175705, 7.33215615418292279930427556593, 7.49138483742766372941795586260, 7.60219208228645671880275015653, 8.049226932103273779527137714584, 8.052697561565362300356543946661, 8.198698155605665546169179274492
