| L(s) = 1 | − 4-s − 3·9-s − 2·11-s + 16-s − 10·19-s − 8·31-s + 3·36-s − 6·41-s + 2·44-s + 14·49-s − 8·59-s − 16·61-s − 64-s − 12·71-s + 10·76-s − 28·79-s + 14·89-s + 6·99-s + 12·101-s + 12·109-s − 19·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 9-s − 0.603·11-s + 1/4·16-s − 2.29·19-s − 1.43·31-s + 1/2·36-s − 0.937·41-s + 0.301·44-s + 2·49-s − 1.04·59-s − 2.04·61-s − 1/8·64-s − 1.42·71-s + 1.14·76-s − 3.15·79-s + 1.48·89-s + 0.603·99-s + 1.19·101-s + 1.14·109-s − 1.72·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04050408865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04050408865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−9.095446741134938975361896430860, −9.000966820342868989578596781248, −8.790818611242418958286584539239, −8.400213925352997832824138565696, −7.80338259406801486371003718628, −7.67557595544474979412347171329, −7.05571225588969988773460255718, −6.70494542988074724774493996844, −6.04709913803846023219198329529, −5.84164120198541479031362987727, −5.57112573333003287900307051115, −4.84875168779007060191989644726, −4.56523500952116029024604501821, −4.13323733935386201006904648718, −3.56622452256058221211025062499, −3.08579888448645632818647788432, −2.52782078586486341379761992060, −2.05492021980020190922706994075, −1.37032365811489866772754692917, −0.07343599865209760623011349203,
0.07343599865209760623011349203, 1.37032365811489866772754692917, 2.05492021980020190922706994075, 2.52782078586486341379761992060, 3.08579888448645632818647788432, 3.56622452256058221211025062499, 4.13323733935386201006904648718, 4.56523500952116029024604501821, 4.84875168779007060191989644726, 5.57112573333003287900307051115, 5.84164120198541479031362987727, 6.04709913803846023219198329529, 6.70494542988074724774493996844, 7.05571225588969988773460255718, 7.67557595544474979412347171329, 7.80338259406801486371003718628, 8.400213925352997832824138565696, 8.790818611242418958286584539239, 9.000966820342868989578596781248, 9.095446741134938975361896430860
