| L(s) = 1 | + 2-s + 5·7-s − 8-s + 12·11-s − 5·13-s + 5·14-s − 16-s − 6·17-s + 4·19-s + 12·22-s + 12·23-s − 10·25-s − 5·26-s − 6·29-s + 31-s − 6·34-s + 37-s + 4·38-s + 6·41-s + 43-s + 12·46-s + 6·47-s + 18·49-s − 10·50-s + 6·53-s − 5·56-s − 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.88·7-s − 0.353·8-s + 3.61·11-s − 1.38·13-s + 1.33·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s + 2.55·22-s + 2.50·23-s − 2·25-s − 0.980·26-s − 1.11·29-s + 0.179·31-s − 1.02·34-s + 0.164·37-s + 0.648·38-s + 0.937·41-s + 0.152·43-s + 1.76·46-s + 0.875·47-s + 18/7·49-s − 1.41·50-s + 0.824·53-s − 0.668·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.869721843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.869721843\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 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
−9.710392721598376877064088641370, −9.465332533031813976941574797422, −9.120305119157853814756542819241, −9.105797657443044987251634530276, −8.382233093431997321324514039656, −8.009311720512961542835125142076, −7.25040745162203603056926161355, −7.19898118594152866756694412444, −6.68477463206211802736074348248, −6.33861633985820331710232424977, −5.43328226005505491157279947663, −5.42275544283082452023052915588, −4.80342790808219005803640306317, −4.25284980915254941516585475812, −3.88168682084895238490402238393, −3.88000364443450822150639819388, −2.67997420454366301469499555766, −2.18130811670038671856949365784, −1.45918747090600352519881053067, −1.00651431091087365133455105355,
1.00651431091087365133455105355, 1.45918747090600352519881053067, 2.18130811670038671856949365784, 2.67997420454366301469499555766, 3.88000364443450822150639819388, 3.88168682084895238490402238393, 4.25284980915254941516585475812, 4.80342790808219005803640306317, 5.42275544283082452023052915588, 5.43328226005505491157279947663, 6.33861633985820331710232424977, 6.68477463206211802736074348248, 7.19898118594152866756694412444, 7.25040745162203603056926161355, 8.009311720512961542835125142076, 8.382233093431997321324514039656, 9.105797657443044987251634530276, 9.120305119157853814756542819241, 9.465332533031813976941574797422, 9.710392721598376877064088641370
