| L(s) = 1 | − 2·5-s + 8·7-s − 9-s + 6·13-s − 25-s + 4·29-s − 16·35-s + 4·37-s + 2·45-s + 24·47-s + 34·49-s + 4·61-s − 8·63-s − 12·65-s − 16·67-s + 28·73-s + 81-s + 48·91-s + 20·97-s + 12·101-s − 6·117-s + 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 3.02·7-s − 1/3·9-s + 1.66·13-s − 1/5·25-s + 0.742·29-s − 2.70·35-s + 0.657·37-s + 0.298·45-s + 3.50·47-s + 34/7·49-s + 0.512·61-s − 1.00·63-s − 1.48·65-s − 1.95·67-s + 3.27·73-s + 1/9·81-s + 5.03·91-s + 2.03·97-s + 1.19·101-s − 0.554·117-s + 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.679718967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.679718967\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.355118847016640669126619370066, −9.169294665954572934141170541735, −8.543257586551509662726399809061, −8.370531563723159054149948428995, −8.193110180435587004069112345124, −7.75368261970496526307225846925, −7.32853514997357475942068547551, −7.13455730396615468720106897032, −6.18857571069283806154243246272, −5.97885244987649186867030418719, −5.49122488156977747994936296102, −4.99517588491925075461682532877, −4.51351829038816300382459756230, −4.35412698572873041115562398450, −3.72879738138337681128420543963, −3.39030434460100676598553488403, −2.26985485437654825399048318217, −2.17406330655496006055003753015, −1.14370376904506975046133358182, −0.963883467554278500786506328594,
0.963883467554278500786506328594, 1.14370376904506975046133358182, 2.17406330655496006055003753015, 2.26985485437654825399048318217, 3.39030434460100676598553488403, 3.72879738138337681128420543963, 4.35412698572873041115562398450, 4.51351829038816300382459756230, 4.99517588491925075461682532877, 5.49122488156977747994936296102, 5.97885244987649186867030418719, 6.18857571069283806154243246272, 7.13455730396615468720106897032, 7.32853514997357475942068547551, 7.75368261970496526307225846925, 8.193110180435587004069112345124, 8.370531563723159054149948428995, 8.543257586551509662726399809061, 9.169294665954572934141170541735, 9.355118847016640669126619370066
