L(s) = 1 | + 4·3-s + 2·5-s − 2·7-s + 6·9-s + 2·11-s − 2·13-s + 8·15-s + 16·19-s − 8·21-s + 6·23-s + 11·25-s − 4·27-s − 10·29-s − 10·31-s + 8·33-s − 4·35-s + 16·37-s − 8·39-s + 14·41-s − 10·43-s + 12·45-s − 2·47-s + 9·49-s + 16·53-s + 4·55-s + 64·57-s − 14·59-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 0.894·5-s − 0.755·7-s + 2·9-s + 0.603·11-s − 0.554·13-s + 2.06·15-s + 3.67·19-s − 1.74·21-s + 1.25·23-s + 11/5·25-s − 0.769·27-s − 1.85·29-s − 1.79·31-s + 1.39·33-s − 0.676·35-s + 2.63·37-s − 1.28·39-s + 2.18·41-s − 1.52·43-s + 1.78·45-s − 0.291·47-s + 9/7·49-s + 2.19·53-s + 0.539·55-s + 8.47·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.405962882\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.405962882\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
good | 5 | $C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 2 T - 5 T^{2} - 10 T^{3} + 4 T^{4} - 10 p T^{5} - 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 13 T^{2} + 10 T^{3} + 124 T^{4} + 10 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 T + T^{2} - 46 T^{3} - 212 T^{4} - 46 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T - 13 T^{2} - 18 T^{3} + 1044 T^{4} - 18 p T^{5} - 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 10 T + 41 T^{2} + 10 T^{3} - 260 T^{4} + 10 p T^{5} + 41 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 10 T + 19 T^{2} + 190 T^{3} + 2500 T^{4} + 190 p T^{5} + 19 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 14 T + 89 T^{2} - 350 T^{3} + 1732 T^{4} - 350 p T^{5} + 89 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 47 | $D_4\times C_2$ | \( 1 + 2 T - 37 T^{2} - 106 T^{3} - 716 T^{4} - 106 p T^{5} - 37 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 14 T + 83 T^{2} - 70 T^{3} - 2276 T^{4} - 70 p T^{5} + 83 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 71 T^{2} + 90 T^{3} + 5532 T^{4} + 90 p T^{5} - 71 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 10 T - 5 T^{2} - 290 T^{3} - 164 T^{4} - 290 p T^{5} - 5 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 22 T + 211 T^{2} + 2530 T^{3} + 30052 T^{4} + 2530 p T^{5} + 211 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 18 p T^{5} - 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 167 T^{2} - 46 T^{3} + 19444 T^{4} - 46 p T^{5} - 167 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.802366849591633104212196581501, −8.384599591567832178479766498476, −8.270616470193019791913938116233, −7.76080480194947032409279009545, −7.40308129828729309200907161350, −7.37718730072852016167645715841, −7.22753108179000511519004744035, −7.05331287263107011571038023680, −6.78469894815488152792174459391, −6.03760891892484447331283984319, −5.76431442007301978530754523789, −5.67280859816382910345445622436, −5.66970187538746219855186958130, −5.10710065684933332127265875231, −4.77378150722055495741520014556, −4.22314448988996876319154296437, −4.12443554814748482409704625008, −3.57442300361465014986418461469, −3.30801015984623912064459607166, −2.97200087401570772926206173816, −2.85807433833906804000361864764, −2.62400567395896209033707731373, −2.07552688324018637090115211045, −1.43111160062968314284468391098, −1.10551250358527615672049100448,
1.10551250358527615672049100448, 1.43111160062968314284468391098, 2.07552688324018637090115211045, 2.62400567395896209033707731373, 2.85807433833906804000361864764, 2.97200087401570772926206173816, 3.30801015984623912064459607166, 3.57442300361465014986418461469, 4.12443554814748482409704625008, 4.22314448988996876319154296437, 4.77378150722055495741520014556, 5.10710065684933332127265875231, 5.66970187538746219855186958130, 5.67280859816382910345445622436, 5.76431442007301978530754523789, 6.03760891892484447331283984319, 6.78469894815488152792174459391, 7.05331287263107011571038023680, 7.22753108179000511519004744035, 7.37718730072852016167645715841, 7.40308129828729309200907161350, 7.76080480194947032409279009545, 8.270616470193019791913938116233, 8.384599591567832178479766498476, 8.802366849591633104212196581501