| L(s) = 1 | + 5·3-s − 3·4-s + 16·9-s − 15·12-s + 20·13-s − 7·16-s + 14·19-s + 35·27-s + 84·31-s − 48·36-s − 80·37-s + 100·39-s − 100·43-s − 35·48-s − 98·49-s − 60·52-s + 70·57-s − 16·61-s + 69·64-s + 90·67-s − 70·73-s − 42·76-s + 24·79-s + 31·81-s + 420·93-s − 140·97-s − 140·103-s + ⋯ |
| L(s) = 1 | + 5/3·3-s − 3/4·4-s + 16/9·9-s − 5/4·12-s + 1.53·13-s − 0.437·16-s + 0.736·19-s + 1.29·27-s + 2.70·31-s − 4/3·36-s − 2.16·37-s + 2.56·39-s − 2.32·43-s − 0.729·48-s − 2·49-s − 1.15·52-s + 1.22·57-s − 0.262·61-s + 1.07·64-s + 1.34·67-s − 0.958·73-s − 0.552·76-s + 0.303·79-s + 0.382·81-s + 4.51·93-s − 1.44·97-s − 1.35·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.180296960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.180296960\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - 5 T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 p T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 567 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 662 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 582 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3087 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2262 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3462 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 45 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8982 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 35 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8927 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 97 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
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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
−14.95953182210514325200891352014, −13.72782853245592877672190168231, −13.55862194197754773092735335042, −13.49279919352567698479479415926, −12.59337311057663701345596592461, −11.91069305517941890141965717447, −11.29199451049496471615975298792, −10.48430964046182217552495818096, −9.766789116580336746736852012413, −9.546116413985088768173499340921, −8.549041823923230332579214224462, −8.492993092380618009236812325576, −8.030741832526778092557111329723, −6.93171704267930231635187998153, −6.46956382172972312094536406113, −5.22302058591136755183603019448, −4.45213285449670508195301003069, −3.59858777141580928655827625422, −3.00064641519853935587648086936, −1.57153493423383635095801869848,
1.57153493423383635095801869848, 3.00064641519853935587648086936, 3.59858777141580928655827625422, 4.45213285449670508195301003069, 5.22302058591136755183603019448, 6.46956382172972312094536406113, 6.93171704267930231635187998153, 8.030741832526778092557111329723, 8.492993092380618009236812325576, 8.549041823923230332579214224462, 9.546116413985088768173499340921, 9.766789116580336746736852012413, 10.48430964046182217552495818096, 11.29199451049496471615975298792, 11.91069305517941890141965717447, 12.59337311057663701345596592461, 13.49279919352567698479479415926, 13.55862194197754773092735335042, 13.72782853245592877672190168231, 14.95953182210514325200891352014
