| L(s) = 1 | + 2·3-s + 3·9-s + 12·17-s + 8·19-s + 2·25-s + 4·27-s + 12·41-s + 8·43-s − 2·49-s + 24·51-s + 16·57-s − 24·59-s − 8·67-s − 4·73-s + 4·75-s + 5·81-s − 12·89-s − 4·97-s − 24·107-s − 12·113-s − 22·121-s + 24·123-s + 127-s + 16·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 2.91·17-s + 1.83·19-s + 2/5·25-s + 0.769·27-s + 1.87·41-s + 1.21·43-s − 2/7·49-s + 3.36·51-s + 2.11·57-s − 3.12·59-s − 0.977·67-s − 0.468·73-s + 0.461·75-s + 5/9·81-s − 1.27·89-s − 0.406·97-s − 2.32·107-s − 1.12·113-s − 2·121-s + 2.16·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.866760827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.866760827\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−10.39521073874020647516114032943, −10.01708505222310147168735736375, −9.465714208982566595750229259561, −9.372667454474063667608579411767, −9.039980779626859621063654855052, −8.214914240670044582931981527415, −7.889096077611923200869152168690, −7.64586829684534494324413764162, −7.35668311255951904780242918563, −6.79706933710280097510253097845, −5.88535346965831285894124540745, −5.79999482200306713163202956010, −5.17519834502749886916405280921, −4.60778667873856736773533259212, −3.94473219590269394985990184396, −3.46641820802356441166979549669, −2.82727781584139604585754955534, −2.79523610059826734126120834496, −1.40376133463709272202370477837, −1.16462166520986921322379175886,
1.16462166520986921322379175886, 1.40376133463709272202370477837, 2.79523610059826734126120834496, 2.82727781584139604585754955534, 3.46641820802356441166979549669, 3.94473219590269394985990184396, 4.60778667873856736773533259212, 5.17519834502749886916405280921, 5.79999482200306713163202956010, 5.88535346965831285894124540745, 6.79706933710280097510253097845, 7.35668311255951904780242918563, 7.64586829684534494324413764162, 7.889096077611923200869152168690, 8.214914240670044582931981527415, 9.039980779626859621063654855052, 9.372667454474063667608579411767, 9.465714208982566595750229259561, 10.01708505222310147168735736375, 10.39521073874020647516114032943
