| L(s) = 1 | + 2·3-s − 5-s + 9-s − 11-s − 10·13-s − 2·15-s − 8·17-s + 5·19-s + 8·23-s − 4·25-s − 2·27-s + 6·29-s + 2·31-s − 2·33-s − 3·37-s − 20·39-s − 12·41-s + 14·43-s − 45-s + 12·47-s − 16·51-s − 11·53-s + 55-s + 10·57-s − 5·59-s + 20·61-s + 10·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 2.77·13-s − 0.516·15-s − 1.94·17-s + 1.14·19-s + 1.66·23-s − 4/5·25-s − 0.384·27-s + 1.11·29-s + 0.359·31-s − 0.348·33-s − 0.493·37-s − 3.20·39-s − 1.87·41-s + 2.13·43-s − 0.149·45-s + 1.75·47-s − 2.24·51-s − 1.51·53-s + 0.134·55-s + 1.32·57-s − 0.650·59-s + 2.56·61-s + 1.24·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.169208946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.169208946\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + T - 7 T^{2} - 14 T^{3} - 68 T^{4} - 14 p T^{5} - 7 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 5 T - 5 T^{2} + 40 T^{3} + 64 T^{4} + 40 p T^{5} - 5 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 3 T - 53 T^{2} - 36 T^{3} + 2142 T^{4} - 36 p T^{5} - 53 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 11 T - T^{2} + 176 T^{3} + 5662 T^{4} + 176 p T^{5} - p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 5 T - 85 T^{2} - 40 T^{3} + 7144 T^{4} - 40 p T^{5} - 85 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 7 T - 83 T^{2} + 14 T^{3} + 9652 T^{4} + 14 p T^{5} - 83 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + T - 131 T^{2} - 14 T^{3} + 12022 T^{4} - 14 p T^{5} - 131 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 8 T - 53 T^{2} + 328 T^{3} + 2392 T^{4} + 328 p T^{5} - 53 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 7 T + 164 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 94 T^{2} - 288 T^{3} + 5775 T^{4} - 288 p T^{5} - 94 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 25 T + 336 T^{2} + 25 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.64064379143275012134430479098, −6.12589913968863992894977274003, −5.89608631285451916582971448154, −5.77516716760922360435652456157, −5.47751711622525871773557786241, −5.23542940510970871024279005429, −5.05152519120248694679734574939, −4.91418258126511507797199753123, −4.71324592271630888377744880141, −4.34738368378736158257136114385, −4.34492976942104251977218069923, −4.14664757588303007830343285264, −3.67930847594439919728008632825, −3.57783262419961320200628409693, −3.16067062382694654740118550749, −3.07005034339211272080342764424, −2.81019490162364160959297187420, −2.50867186705510096959058133403, −2.39802366218911424416466655087, −2.12285733087057537993755374554, −2.08236311795215321119045462397, −1.38066338343555960813660473771, −1.17463120821293284080853663218, −0.64162726580858511342487884369, −0.19304934816264362709072944297,
0.19304934816264362709072944297, 0.64162726580858511342487884369, 1.17463120821293284080853663218, 1.38066338343555960813660473771, 2.08236311795215321119045462397, 2.12285733087057537993755374554, 2.39802366218911424416466655087, 2.50867186705510096959058133403, 2.81019490162364160959297187420, 3.07005034339211272080342764424, 3.16067062382694654740118550749, 3.57783262419961320200628409693, 3.67930847594439919728008632825, 4.14664757588303007830343285264, 4.34492976942104251977218069923, 4.34738368378736158257136114385, 4.71324592271630888377744880141, 4.91418258126511507797199753123, 5.05152519120248694679734574939, 5.23542940510970871024279005429, 5.47751711622525871773557786241, 5.77516716760922360435652456157, 5.89608631285451916582971448154, 6.12589913968863992894977274003, 6.64064379143275012134430479098
