| L(s) = 1 | + 8·11-s − 4·13-s − 8·23-s + 8·25-s − 16·37-s + 24·47-s + 6·49-s − 16·61-s − 8·71-s − 16·73-s − 24·83-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 1.10·13-s − 1.66·23-s + 8/5·25-s − 2.63·37-s + 3.50·47-s + 6/7·49-s − 2.04·61-s − 0.949·71-s − 1.87·73-s − 2.63·83-s − 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.141090536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.141090536\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.050764174329470876646657473402, −8.854456788050235105825443435983, −8.756675239228688077477295207637, −7.979830756310793434057763303183, −7.64565204342405753124134377933, −7.03029615225512513361553040965, −6.96849944292157720317663448789, −6.68402599827655356248212134384, −6.02564979351818028226375934443, −5.61867582369389244808268437133, −5.47449318781967526696586743128, −4.57084023428037797920991934121, −4.34355474974664009810896377217, −4.06153085615026948696850569913, −3.53542502115651030832999939593, −2.92602808677589458751461621181, −2.51938436368516281918204318560, −1.56940360901869593413725878476, −1.55868740680514076662267165274, −0.50927834560526977543025369583,
0.50927834560526977543025369583, 1.55868740680514076662267165274, 1.56940360901869593413725878476, 2.51938436368516281918204318560, 2.92602808677589458751461621181, 3.53542502115651030832999939593, 4.06153085615026948696850569913, 4.34355474974664009810896377217, 4.57084023428037797920991934121, 5.47449318781967526696586743128, 5.61867582369389244808268437133, 6.02564979351818028226375934443, 6.68402599827655356248212134384, 6.96849944292157720317663448789, 7.03029615225512513361553040965, 7.64565204342405753124134377933, 7.979830756310793434057763303183, 8.756675239228688077477295207637, 8.854456788050235105825443435983, 9.050764174329470876646657473402
