| L(s) = 1 | + 2·2-s + 2·4-s − 8·5-s − 12·7-s + 4·8-s − 16·10-s − 4·11-s − 24·14-s + 8·16-s + 6·17-s + 6·19-s − 16·20-s − 8·22-s + 6·23-s + 26·25-s − 24·28-s + 8·32-s + 12·34-s + 96·35-s + 12·37-s + 12·38-s − 32·40-s + 18·41-s − 12·43-s − 8·44-s + 12·46-s + 70·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 3.57·5-s − 4.53·7-s + 1.41·8-s − 5.05·10-s − 1.20·11-s − 6.41·14-s + 2·16-s + 1.45·17-s + 1.37·19-s − 3.57·20-s − 1.70·22-s + 1.25·23-s + 26/5·25-s − 4.53·28-s + 1.41·32-s + 2.05·34-s + 16.2·35-s + 1.97·37-s + 1.94·38-s − 5.05·40-s + 2.81·41-s − 1.82·43-s − 1.20·44-s + 1.76·46-s + 10·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8195449897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8195449897\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + 5 T^{2} + 18 T^{3} + 60 T^{4} + 18 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T - 8 T^{2} - 36 T^{3} + 891 T^{4} - 36 p T^{5} - 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T + 8 T^{2} + 108 T^{3} - 573 T^{4} + 108 p T^{5} + 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 49 T^{2} + 1560 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4314 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_4\times C_2$ | \( 1 - 18 T + 201 T^{2} - 1674 T^{3} + 11396 T^{4} - 1674 p T^{5} + 201 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 12 T + 110 T^{2} + 744 T^{3} + 4059 T^{4} + 744 p T^{5} + 110 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 98 T^{2} + 6291 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T - 58 T^{2} + 32 T^{3} + 7627 T^{4} + 32 p T^{5} - 58 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 47 T^{2} + p^{2} T^{4} )( 1 + 74 T^{2} + p^{2} T^{4} ) \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 112 T^{2} - 1404 T^{3} + 18747 T^{4} - 1404 p T^{5} + 112 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 36 T^{2} - 144 T^{3} - 3613 T^{4} - 144 p T^{5} + 36 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 158 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 222 T^{2} + 2088 T^{3} + 26627 T^{4} + 2088 p T^{5} + 222 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.14932147757115709517479286351, −7.00396500620774179454701122414, −6.86694200402559778225905857166, −6.51046030466102754969834702749, −6.39460065824097365589405066876, −5.94327190136883027051425537831, −5.93555087171879829135817763144, −5.45471719228046036740070047748, −5.44258615311702612758194650688, −5.21106432271597557731762028173, −4.68556433394280365489396628863, −4.37780148477006967609613668145, −4.35373340253496416742980542653, −4.06390299523671024597746831004, −3.74192087355765304226224865282, −3.45630077140752615472690908528, −3.45167755170296340760670696413, −3.29602147932287314197795783424, −2.98344056565803552958265904600, −2.75267072976103626288913748477, −2.67890572817001675659794805249, −1.89887000555132024580175752835, −0.861396753172196908325327668250, −0.78726659694127066295922914397, −0.29352901180768094508545508622,
0.29352901180768094508545508622, 0.78726659694127066295922914397, 0.861396753172196908325327668250, 1.89887000555132024580175752835, 2.67890572817001675659794805249, 2.75267072976103626288913748477, 2.98344056565803552958265904600, 3.29602147932287314197795783424, 3.45167755170296340760670696413, 3.45630077140752615472690908528, 3.74192087355765304226224865282, 4.06390299523671024597746831004, 4.35373340253496416742980542653, 4.37780148477006967609613668145, 4.68556433394280365489396628863, 5.21106432271597557731762028173, 5.44258615311702612758194650688, 5.45471719228046036740070047748, 5.93555087171879829135817763144, 5.94327190136883027051425537831, 6.39460065824097365589405066876, 6.51046030466102754969834702749, 6.86694200402559778225905857166, 7.00396500620774179454701122414, 7.14932147757115709517479286351
