| L(s) = 1 | − 8·7-s − 16·13-s + 64·19-s − 32·25-s + 88·31-s − 136·37-s − 80·43-s + 114·49-s − 100·61-s + 16·67-s − 64·73-s − 152·79-s + 128·91-s − 352·97-s − 56·103-s + 46·121-s + 127-s + 131-s − 512·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 402·169-s + ⋯ |
| L(s) = 1 | − 8/7·7-s − 1.23·13-s + 3.36·19-s − 1.27·25-s + 2.83·31-s − 3.67·37-s − 1.86·43-s + 2.32·49-s − 1.63·61-s + 0.238·67-s − 0.876·73-s − 1.92·79-s + 1.40·91-s − 3.62·97-s − 0.543·103-s + 0.380·121-s + 0.00787·127-s + 0.00763·131-s − 3.84·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.37·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.590879889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.590879889\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 32 T^{2} + 399 T^{4} + 32 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 4 T - 33 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 75 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )( 1 + 14 T + 75 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T - 105 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 416 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 770 T^{2} + 313059 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 1664 T^{2} + 2061615 T^{4} + 1664 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 44 T + 975 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 1184 T^{2} - 1423905 T^{4} + 1184 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 40 T - 249 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 2782 T^{2} + 2859843 T^{4} - 2782 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 4160 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 5810 T^{2} + 21638739 T^{4} + 5810 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 50 T - 1221 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 8 T - 4425 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7490 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 76 T - 465 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 334 T^{2} - 47346765 T^{4} - 334 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 15680 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 176 T + 21567 T^{2} + 176 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.73383323466239216545292087952, −6.68159005841675734277204337448, −6.13214046530207413774425331884, −6.02118810136560461314643480517, −5.82077498142723927320191538175, −5.55465288284498307877700565553, −5.47233813852898855140175764403, −5.06154193222875592479225219724, −4.91356047486478775495262055357, −4.78235955372199031578169960361, −4.57568465772387595774941140746, −4.04649999288718766986586578256, −3.94984171510819777066403177691, −3.52514491224978052725484831343, −3.39676378052576679278060865521, −3.27777401816635810601195933565, −2.75254598387340981625669053704, −2.72884802537114824697392464498, −2.59348341771777709382306834740, −2.03142903507950591933560985407, −1.57043179069991793507059173227, −1.31350147956933695268378227765, −1.21628897945081299259947134139, −0.42598973697715880046712324548, −0.27191087641061814251547865218,
0.27191087641061814251547865218, 0.42598973697715880046712324548, 1.21628897945081299259947134139, 1.31350147956933695268378227765, 1.57043179069991793507059173227, 2.03142903507950591933560985407, 2.59348341771777709382306834740, 2.72884802537114824697392464498, 2.75254598387340981625669053704, 3.27777401816635810601195933565, 3.39676378052576679278060865521, 3.52514491224978052725484831343, 3.94984171510819777066403177691, 4.04649999288718766986586578256, 4.57568465772387595774941140746, 4.78235955372199031578169960361, 4.91356047486478775495262055357, 5.06154193222875592479225219724, 5.47233813852898855140175764403, 5.55465288284498307877700565553, 5.82077498142723927320191538175, 6.02118810136560461314643480517, 6.13214046530207413774425331884, 6.68159005841675734277204337448, 6.73383323466239216545292087952
