| L(s) = 1 | − 2·5-s + 6·9-s − 6·13-s + 16·17-s + 11·25-s + 2·29-s + 32·37-s − 10·41-s − 12·45-s + 11·49-s + 32·53-s − 14·61-s + 12·65-s − 48·73-s + 27·81-s − 32·85-s − 16·89-s + 6·97-s − 26·101-s + 2·113-s − 36·117-s + 19·121-s − 38·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2·9-s − 1.66·13-s + 3.88·17-s + 11/5·25-s + 0.371·29-s + 5.26·37-s − 1.56·41-s − 1.78·45-s + 11/7·49-s + 4.39·53-s − 1.79·61-s + 1.48·65-s − 5.61·73-s + 3·81-s − 3.47·85-s − 1.69·89-s + 0.609·97-s − 2.58·101-s + 0.188·113-s − 3.32·117-s + 1.72·121-s − 3.39·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.628286680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.628286680\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - 19 T^{2} + 240 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 29 T^{2} + 312 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 35 T^{2} + 264 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 11 T^{2} - 1728 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 53 T^{2} + 600 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 59 T^{2} - 1008 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 83 | $C_2^3$ | \( 1 - 91 T^{2} + 1392 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 3 T - 88 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.68473039817180142527553158022, −7.42316762803947496260423256939, −7.37590504463993109679122082694, −7.20537508240674074125329787778, −7.01020004229341744030646703982, −6.62946937773544826787486948843, −6.17815024321373516524761835285, −6.10719887708062703033876344820, −5.82632683410062399018216428840, −5.42345236604146557386579776974, −5.23770239913816992599115931990, −5.04661427229553903000466013317, −4.77990081670115806645181361534, −4.23725515844151126094460315478, −4.17145312869227805869359197685, −4.08702260743583094319812045499, −3.87577459947347825383008362617, −3.15688185725930007471560628015, −2.88473684136154877022035332505, −2.78894453054846726077968288434, −2.64210390742601248164174795270, −1.84892681640977645116623063046, −1.21157499120624372355627940872, −1.14955215450611724539340734121, −0.74824771779610708114160593768,
0.74824771779610708114160593768, 1.14955215450611724539340734121, 1.21157499120624372355627940872, 1.84892681640977645116623063046, 2.64210390742601248164174795270, 2.78894453054846726077968288434, 2.88473684136154877022035332505, 3.15688185725930007471560628015, 3.87577459947347825383008362617, 4.08702260743583094319812045499, 4.17145312869227805869359197685, 4.23725515844151126094460315478, 4.77990081670115806645181361534, 5.04661427229553903000466013317, 5.23770239913816992599115931990, 5.42345236604146557386579776974, 5.82632683410062399018216428840, 6.10719887708062703033876344820, 6.17815024321373516524761835285, 6.62946937773544826787486948843, 7.01020004229341744030646703982, 7.20537508240674074125329787778, 7.37590504463993109679122082694, 7.42316762803947496260423256939, 7.68473039817180142527553158022
