| L(s) = 1 | − 8·5-s + 18·9-s − 48·13-s + 39·25-s + 48·37-s + 36·41-s − 144·45-s − 98·49-s − 112·53-s + 384·65-s + 243·81-s + 156·89-s − 864·117-s − 242·121-s − 112·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.39e3·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 8/5·5-s + 2·9-s − 3.69·13-s + 1.55·25-s + 1.29·37-s + 0.878·41-s − 3.19·45-s − 2·49-s − 2.11·53-s + 5.90·65-s + 3·81-s + 1.75·89-s − 7.38·117-s − 2·121-s − 0.895·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 8.22·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4915197193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4915197193\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 8 T + p^{2} T^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )( 1 + 40 T + p^{2} T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )( 1 + 120 T + p^{2} T^{2} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )( 1 + 110 T + p^{2} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 78 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )( 1 + 130 T + p^{2} T^{2} ) \) |
| 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
−9.584539920839557319699422119091, −9.520586790995276151696775267170, −9.048928183708732827323776216092, −8.162893297360261222909411995941, −7.82971637719007988572558348077, −7.62685543525819507605177862884, −7.41863929300366455259484011790, −6.94352402750425767189654553222, −6.63781107990241485175812735820, −6.07240497627949880392792874209, −5.05697853509643276692118730466, −4.79821741029783324008736263543, −4.72381054953396964295861719865, −4.17818707283162446341061340425, −3.70538854698344814360187314448, −3.02867791339672783882452918539, −2.51348918232917226017457985967, −1.94356092839661829544362303932, −1.10379369077931949583608804204, −0.22622255077772809744827985490,
0.22622255077772809744827985490, 1.10379369077931949583608804204, 1.94356092839661829544362303932, 2.51348918232917226017457985967, 3.02867791339672783882452918539, 3.70538854698344814360187314448, 4.17818707283162446341061340425, 4.72381054953396964295861719865, 4.79821741029783324008736263543, 5.05697853509643276692118730466, 6.07240497627949880392792874209, 6.63781107990241485175812735820, 6.94352402750425767189654553222, 7.41863929300366455259484011790, 7.62685543525819507605177862884, 7.82971637719007988572558348077, 8.162893297360261222909411995941, 9.048928183708732827323776216092, 9.520586790995276151696775267170, 9.584539920839557319699422119091
