| L(s) = 1 | − 8·2-s − 14·3-s + 48·4-s + 112·6-s − 256·8-s + 81·9-s + 46·11-s − 672·12-s + 1.28e3·16-s − 574·17-s − 648·18-s − 434·19-s − 368·22-s + 3.58e3·24-s − 658·27-s − 6.14e3·32-s − 644·33-s + 4.59e3·34-s + 3.88e3·36-s + 3.47e3·38-s + 1.24e3·41-s + 7.00e3·43-s + 2.20e3·44-s − 1.79e4·48-s + 4.80e3·49-s + 8.03e3·51-s + 5.26e3·54-s + ⋯ |
| L(s) = 1 | − 2·2-s − 1.55·3-s + 3·4-s + 28/9·6-s − 4·8-s + 9-s + 0.380·11-s − 4.66·12-s + 5·16-s − 1.98·17-s − 2·18-s − 1.20·19-s − 0.760·22-s + 56/9·24-s − 0.902·27-s − 6·32-s − 0.591·33-s + 3.97·34-s + 3·36-s + 2.40·38-s + 0.741·41-s + 3.78·43-s + 1.14·44-s − 7.77·48-s + 2·49-s + 3.08·51-s + 1.80·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3118441225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3118441225\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 14 T + 115 T^{2} + 14 p^{4} T^{3} + p^{8} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 46 T - 12525 T^{2} - 46 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 574 T + 245955 T^{2} + 574 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 434 T + 58035 T^{2} + 434 p^{4} T^{3} + p^{8} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1246 T - 1273245 T^{2} - 1246 p^{4} T^{3} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 3502 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 238 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 5134 T + 6206835 T^{2} + 5134 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 9506 T + 61965795 T^{2} - 9506 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 11186 T + 77668275 T^{2} - 11186 p^{4} T^{3} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 5474 T - 32777565 T^{2} + 5474 p^{4} T^{3} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 9982 T + p^{4} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.55852343339887719651891746500, −11.46015053702865324138090867972, −10.86996835469171519178175250156, −10.58633758839248260786160592519, −10.40032049911178045407657506635, −9.323674867810153077182779290566, −9.108777141990701022901729373663, −8.865875741756638092884951730610, −7.913869875137284444252864711658, −7.57166602161902719632328263542, −6.79389526969641658467963002185, −6.57602513123996411529280143727, −5.83717573269861283910582950020, −5.76584857116866700343073870279, −4.53714730575734771522536165695, −3.85810897581730552382728828566, −2.44554982505558064006544382926, −2.13408805326606852900954787862, −0.923441901973232581586968420286, −0.39167891092237386639538168271,
0.39167891092237386639538168271, 0.923441901973232581586968420286, 2.13408805326606852900954787862, 2.44554982505558064006544382926, 3.85810897581730552382728828566, 4.53714730575734771522536165695, 5.76584857116866700343073870279, 5.83717573269861283910582950020, 6.57602513123996411529280143727, 6.79389526969641658467963002185, 7.57166602161902719632328263542, 7.913869875137284444252864711658, 8.865875741756638092884951730610, 9.108777141990701022901729373663, 9.323674867810153077182779290566, 10.40032049911178045407657506635, 10.58633758839248260786160592519, 10.86996835469171519178175250156, 11.46015053702865324138090867972, 11.55852343339887719651891746500
