| L(s) = 1 | + 6·3-s + 27·9-s + 22·13-s − 50·25-s + 108·27-s + 132·39-s − 44·43-s − 94·49-s − 148·61-s − 300·75-s + 284·79-s + 405·81-s + 388·103-s + 594·117-s − 242·121-s + 127-s − 264·129-s + 131-s + 137-s + 139-s − 564·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 315·169-s + ⋯ |
| L(s) = 1 | + 2·3-s + 3·9-s + 1.69·13-s − 2·25-s + 4·27-s + 3.38·39-s − 1.02·43-s − 1.91·49-s − 2.42·61-s − 4·75-s + 3.59·79-s + 5·81-s + 3.76·103-s + 5.07·117-s − 2·121-s + 0.00787·127-s − 2.04·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 3.83·147-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.86·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.069620948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.069620948\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_2$ | \( 1 - 22 T + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 26 T + p^{2} T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 26 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )( 1 + 122 T + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
| 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
−13.02137175110910896803892903651, −12.83027034397874242794374695883, −11.98785709953256124230032777807, −11.52181901803581901807449642755, −10.76159758831847064295747210380, −10.33208687075657886424760164615, −9.726425204815955365767502640348, −9.270317286901493890691057848516, −8.867330101408933992957035397080, −8.259592189809295024367060192218, −7.82820186631401468926714571925, −7.54792676377688128567911589748, −6.42129441290886595592647160121, −6.32480416632166768017789165149, −5.08101000913152074298920207679, −4.29441867999210546013038980781, −3.57789698065244507171580911664, −3.29743897817722413101180926532, −2.14471798097812147323850553877, −1.45393196814857371145266615037,
1.45393196814857371145266615037, 2.14471798097812147323850553877, 3.29743897817722413101180926532, 3.57789698065244507171580911664, 4.29441867999210546013038980781, 5.08101000913152074298920207679, 6.32480416632166768017789165149, 6.42129441290886595592647160121, 7.54792676377688128567911589748, 7.82820186631401468926714571925, 8.259592189809295024367060192218, 8.867330101408933992957035397080, 9.270317286901493890691057848516, 9.726425204815955365767502640348, 10.33208687075657886424760164615, 10.76159758831847064295747210380, 11.52181901803581901807449642755, 11.98785709953256124230032777807, 12.83027034397874242794374695883, 13.02137175110910896803892903651
