| L(s) = 1 | − 20·5-s + 6·9-s + 250·25-s + 40·29-s − 136·41-s − 120·45-s − 100·49-s + 280·61-s + 27·81-s − 56·89-s − 152·101-s + 664·109-s + 100·121-s − 2.50e3·125-s + 127-s + 131-s + 137-s + 139-s − 800·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 476·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 4·5-s + 2/3·9-s + 10·25-s + 1.37·29-s − 3.31·41-s − 8/3·45-s − 2.04·49-s + 4.59·61-s + 1/3·81-s − 0.629·89-s − 1.50·101-s + 6.09·109-s + 0.826·121-s − 20·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 5.51·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.81·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9202428371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9202428371\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2}( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2}( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 1730 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2}( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 3266 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4370 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 3314 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6770 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 866 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7010 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 8354 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 10754 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 5666 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 9602 T^{2} + p^{4} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.360749906418505047838562284666, −8.311415037042425722110129273120, −8.212162270013534552609569121276, −7.956726747835421754676356891465, −7.57431927945968984387644451102, −7.16862734300994808733988595190, −7.06080848304232993813917402938, −7.01670312940128071021899464886, −6.59848215050328086432955099877, −6.52004148627212100388231351397, −5.93973573692471481850988484703, −5.36431970002870372659008372572, −5.11057163567614079567434795130, −4.70790685893848110517086853055, −4.62955903193522964836535192408, −4.48673907745250121837261403761, −3.77608954562206136803403845593, −3.67899639709782197662262748747, −3.61432542854607704245833358546, −3.06675403520346530971455154133, −2.90342375248369253056196526615, −2.14602227401143960124045000904, −1.42357051524213698231789824885, −0.70767964702080034427914363184, −0.42683804608392675375214690442,
0.42683804608392675375214690442, 0.70767964702080034427914363184, 1.42357051524213698231789824885, 2.14602227401143960124045000904, 2.90342375248369253056196526615, 3.06675403520346530971455154133, 3.61432542854607704245833358546, 3.67899639709782197662262748747, 3.77608954562206136803403845593, 4.48673907745250121837261403761, 4.62955903193522964836535192408, 4.70790685893848110517086853055, 5.11057163567614079567434795130, 5.36431970002870372659008372572, 5.93973573692471481850988484703, 6.52004148627212100388231351397, 6.59848215050328086432955099877, 7.01670312940128071021899464886, 7.06080848304232993813917402938, 7.16862734300994808733988595190, 7.57431927945968984387644451102, 7.956726747835421754676356891465, 8.212162270013534552609569121276, 8.311415037042425722110129273120, 8.360749906418505047838562284666
