| L(s) = 1 | + 2·2-s + 3-s − 4-s + 2·6-s − 8·8-s + 9-s − 12-s − 7·16-s + 2·18-s + 4·19-s − 8·24-s + 25-s + 27-s + 4·29-s + 14·32-s − 36-s + 8·38-s − 20·41-s + 8·43-s − 7·48-s − 14·49-s + 2·50-s + 20·53-s + 2·54-s + 4·57-s + 8·58-s + 8·59-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s − 1/2·4-s + 0.816·6-s − 2.82·8-s + 1/3·9-s − 0.288·12-s − 7/4·16-s + 0.471·18-s + 0.917·19-s − 1.63·24-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 2.47·32-s − 1/6·36-s + 1.29·38-s − 3.12·41-s + 1.21·43-s − 1.01·48-s − 2·49-s + 0.282·50-s + 2.74·53-s + 0.272·54-s + 0.529·57-s + 1.05·58-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.699915346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.699915346\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.841902196926425418503906012487, −8.554588868470699894187777077269, −8.179427569545746398797779230432, −7.60719004255564017514118173943, −6.76955885158912340897756950945, −6.56470169945492021349835720384, −5.84128873133936523436206316749, −5.21484715703970999635976715829, −5.04860090093668311608273372623, −4.52985320583773135666017839183, −3.66793113681489031905327880745, −3.63412818236731287070311550143, −2.96337185591461309202736907725, −2.19847936643903163153856477125, −0.803857922530386741108529528003,
0.803857922530386741108529528003, 2.19847936643903163153856477125, 2.96337185591461309202736907725, 3.63412818236731287070311550143, 3.66793113681489031905327880745, 4.52985320583773135666017839183, 5.04860090093668311608273372623, 5.21484715703970999635976715829, 5.84128873133936523436206316749, 6.56470169945492021349835720384, 6.76955885158912340897756950945, 7.60719004255564017514118173943, 8.179427569545746398797779230432, 8.554588868470699894187777077269, 8.841902196926425418503906012487
