| L(s) = 1 | − 4·7-s + 16·25-s − 8·31-s + 10·49-s + 8·73-s − 32·79-s − 40·97-s − 8·103-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s − 64·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 16/5·25-s − 1.43·31-s + 10/7·49-s + 0.936·73-s − 3.60·79-s − 4.06·97-s − 0.788·103-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s − 4.83·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.227349592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.227349592\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 88 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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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
−6.58536751853011287508945698470, −6.40960560256093173609281594153, −6.03202868093623232718556537276, −5.87674414793619060584161703313, −5.61177422377085854827023714736, −5.53207119284179617331229228328, −5.33495730075453219098871739664, −5.05690959542482317389938176138, −4.76154428655574153477434967819, −4.57538267370604406321352022439, −4.22461607978317648685925505744, −4.19780101534574803801951816363, −3.96656086446090112757896809189, −3.53028228423401707513150077409, −3.25782221601397802696746710155, −3.22408843350624764201081626539, −2.83637069956899764686076465372, −2.81720901900542935236447918357, −2.55533864782988482486432603834, −2.06929373097144701808234475787, −1.71730875049554199844464589202, −1.56062030942366602967060026466, −0.970350228635461915350124166896, −0.794135502831444623174037573955, −0.22834229170535597974059400605,
0.22834229170535597974059400605, 0.794135502831444623174037573955, 0.970350228635461915350124166896, 1.56062030942366602967060026466, 1.71730875049554199844464589202, 2.06929373097144701808234475787, 2.55533864782988482486432603834, 2.81720901900542935236447918357, 2.83637069956899764686076465372, 3.22408843350624764201081626539, 3.25782221601397802696746710155, 3.53028228423401707513150077409, 3.96656086446090112757896809189, 4.19780101534574803801951816363, 4.22461607978317648685925505744, 4.57538267370604406321352022439, 4.76154428655574153477434967819, 5.05690959542482317389938176138, 5.33495730075453219098871739664, 5.53207119284179617331229228328, 5.61177422377085854827023714736, 5.87674414793619060584161703313, 6.03202868093623232718556537276, 6.40960560256093173609281594153, 6.58536751853011287508945698470
