| L(s) = 1 | − 6·3-s − 6·5-s + 15·9-s − 4·11-s + 36·15-s + 4·17-s − 2·19-s − 18·23-s + 12·25-s − 18·27-s − 6·29-s + 12·31-s + 24·33-s − 12·37-s + 6·41-s − 90·45-s − 30·47-s + 4·49-s − 24·51-s + 12·53-s + 24·55-s + 12·57-s − 18·59-s + 18·61-s − 6·67-s + 108·69-s − 12·71-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 2.68·5-s + 5·9-s − 1.20·11-s + 9.29·15-s + 0.970·17-s − 0.458·19-s − 3.75·23-s + 12/5·25-s − 3.46·27-s − 1.11·29-s + 2.15·31-s + 4.17·33-s − 1.97·37-s + 0.937·41-s − 13.4·45-s − 4.37·47-s + 4/7·49-s − 3.36·51-s + 1.64·53-s + 3.23·55-s + 1.58·57-s − 2.34·59-s + 2.30·61-s − 0.733·67-s + 13.0·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 72 T^{3} + 179 T^{4} + 72 p T^{5} + 24 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 10 T^{2} + 32 T^{3} + 115 T^{4} + 32 p T^{5} - 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 18 T + 180 T^{2} + 1296 T^{3} + 7139 T^{4} + 1296 p T^{5} + 180 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T - 4 T^{2} - 108 T^{3} - 285 T^{4} - 108 p T^{5} - 4 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 6 T + 80 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T + 61 T^{2} - 294 T^{3} + 1212 T^{4} - 294 p T^{5} + 61 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^3$ | \( 1 - 74 T^{2} + 3627 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 30 T + 468 T^{2} + 5040 T^{3} + 40115 T^{4} + 5040 p T^{5} + 468 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 14 T^{2} - 288 T^{3} + 6459 T^{4} - 288 p T^{5} + 14 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 217 T^{2} + 1962 T^{3} + 14772 T^{4} + 1962 p T^{5} + 217 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 18 T + 208 T^{2} - 1800 T^{3} + 12867 T^{4} - 1800 p T^{5} + 208 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T + 113 T^{2} + 606 T^{3} + 6516 T^{4} + 606 p T^{5} + 113 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 12 T - 22 T^{2} + 288 T^{3} + 12291 T^{4} + 288 p T^{5} - 22 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 5 T - 48 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 12 T + 58 T^{2} - 864 T^{3} - 10221 T^{4} - 864 p T^{5} + 58 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 24 T + 382 T^{2} + 4560 T^{3} + 45267 T^{4} + 4560 p T^{5} + 382 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 6 T + 173 T^{2} + 966 T^{3} + 17676 T^{4} + 966 p T^{5} + 173 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.47764733621339253762428209427, −7.09846534150064078162454146164, −6.79268612336515928206921454207, −6.77523702346880101930218217442, −6.36297077635708366275617402124, −6.19103081538373663617802062372, −6.03074170972791034312891217699, −5.88371764144996900914480826913, −5.53950627704779727473066817125, −5.47472899284056516914550446967, −5.26383131420191214725056087745, −4.99830295761809621610591852870, −4.87985188246000690233134908678, −4.31253683958934029031937233432, −4.22563856521221089375876907575, −4.16279995631514366787115659817, −3.95040392874279238336211538598, −3.49486431902112893523638122725, −3.43498210187063801059926520658, −2.85137791666403056010258401361, −2.76398728565106356709405996052, −2.25620591739892058766358954931, −1.70730626978912369534272516663, −1.49939994273791059976384768571, −0.922295206936550637962057402849, 0, 0, 0, 0,
0.922295206936550637962057402849, 1.49939994273791059976384768571, 1.70730626978912369534272516663, 2.25620591739892058766358954931, 2.76398728565106356709405996052, 2.85137791666403056010258401361, 3.43498210187063801059926520658, 3.49486431902112893523638122725, 3.95040392874279238336211538598, 4.16279995631514366787115659817, 4.22563856521221089375876907575, 4.31253683958934029031937233432, 4.87985188246000690233134908678, 4.99830295761809621610591852870, 5.26383131420191214725056087745, 5.47472899284056516914550446967, 5.53950627704779727473066817125, 5.88371764144996900914480826913, 6.03074170972791034312891217699, 6.19103081538373663617802062372, 6.36297077635708366275617402124, 6.77523702346880101930218217442, 6.79268612336515928206921454207, 7.09846534150064078162454146164, 7.47764733621339253762428209427
