| L(s) = 1 | − 2·4-s + 9-s + 16-s − 2·36-s + 2·64-s − 4·79-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2·4-s + 9-s + 16-s − 2·36-s + 2·64-s − 4·79-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6302967171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6302967171\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.32181177316750204734320762764, −5.90451412257362294042994342512, −5.79672324053475848840061591292, −5.49494456103752256754041698531, −5.49483563171297673495130064646, −5.11463551762738628865296784253, −4.92543970175945260740955435605, −4.77580967830460677458052028817, −4.67969709517985621770026115620, −4.36988783741455222517272017635, −4.15967593893003488904474067750, −4.14509172404348767709188493748, −3.85837619821449973749196039083, −3.73752328725421659897846026453, −3.43273239751555506297278398309, −2.96176616992779338001518320308, −2.92387641823813717208499251324, −2.86706591115734972858905404922, −2.30439102216675380266862881058, −2.01247329896486790040110599672, −1.83573580210360279505733256308, −1.58854842116750095714050760257, −1.11557228854828374353590489623, −0.892901649175799724836015149968, −0.37646360222908505327163146353,
0.37646360222908505327163146353, 0.892901649175799724836015149968, 1.11557228854828374353590489623, 1.58854842116750095714050760257, 1.83573580210360279505733256308, 2.01247329896486790040110599672, 2.30439102216675380266862881058, 2.86706591115734972858905404922, 2.92387641823813717208499251324, 2.96176616992779338001518320308, 3.43273239751555506297278398309, 3.73752328725421659897846026453, 3.85837619821449973749196039083, 4.14509172404348767709188493748, 4.15967593893003488904474067750, 4.36988783741455222517272017635, 4.67969709517985621770026115620, 4.77580967830460677458052028817, 4.92543970175945260740955435605, 5.11463551762738628865296784253, 5.49483563171297673495130064646, 5.49494456103752256754041698531, 5.79672324053475848840061591292, 5.90451412257362294042994342512, 6.32181177316750204734320762764
