| L(s) = 1 | − 12·5-s − 28·13-s + 12·17-s + 58·25-s − 60·29-s + 52·37-s + 108·41-s + 50·49-s + 36·53-s − 140·61-s + 336·65-s + 164·73-s − 144·85-s − 228·89-s + 68·97-s + 36·101-s + 68·109-s + 156·113-s − 190·121-s + 36·125-s + 127-s + 131-s + 137-s + 139-s + 720·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.39·5-s − 2.15·13-s + 0.705·17-s + 2.31·25-s − 2.06·29-s + 1.40·37-s + 2.63·41-s + 1.02·49-s + 0.679·53-s − 2.29·61-s + 5.16·65-s + 2.24·73-s − 1.69·85-s − 2.56·89-s + 0.701·97-s + 0.356·101-s + 0.623·109-s + 1.38·113-s − 1.57·121-s + 0.287·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 4.96·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5697073142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5697073142\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 674 T^{2} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1490 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3266 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2690 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6530 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 4894 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 3170 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6674 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13346 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 114 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.75345144727858262592301680257, −12.44265924860360542843150338481, −12.23165220885670224821298281047, −11.46601200249298672302700736578, −11.38259716656164813518973650823, −10.77217194177041724575668979415, −10.03408728744051154472956502973, −9.403956603364523077222145155026, −9.092673828551467846628163152345, −8.011670520374580304986637731104, −7.76425616417001139138464534504, −7.47803541519189779948451965969, −7.05969010944775753451358909286, −5.94119600680682944425000894150, −5.27632234860316317286891936066, −4.32292189886565424927391384033, −4.16836055922499031306811378217, −3.27498344094709693575513037536, −2.39802751389393610322492802237, −0.48689423342634248387221944664,
0.48689423342634248387221944664, 2.39802751389393610322492802237, 3.27498344094709693575513037536, 4.16836055922499031306811378217, 4.32292189886565424927391384033, 5.27632234860316317286891936066, 5.94119600680682944425000894150, 7.05969010944775753451358909286, 7.47803541519189779948451965969, 7.76425616417001139138464534504, 8.011670520374580304986637731104, 9.092673828551467846628163152345, 9.403956603364523077222145155026, 10.03408728744051154472956502973, 10.77217194177041724575668979415, 11.38259716656164813518973650823, 11.46601200249298672302700736578, 12.23165220885670224821298281047, 12.44265924860360542843150338481, 12.75345144727858262592301680257
