| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 9-s − 2·13-s + 5·16-s − 2·17-s − 2·18-s + 2·25-s + 4·26-s − 6·32-s + 4·34-s + 3·36-s − 4·50-s − 6·52-s − 2·53-s + 7·64-s − 6·68-s − 4·72-s − 2·89-s + 6·100-s + 4·101-s + 8·104-s + 4·106-s − 2·117-s + 121-s + 127-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 9-s − 2·13-s + 5·16-s − 2·17-s − 2·18-s + 2·25-s + 4·26-s − 6·32-s + 4·34-s + 3·36-s − 4·50-s − 6·52-s − 2·53-s + 7·64-s − 6·68-s − 4·72-s − 2·89-s + 6·100-s + 4·101-s + 8·104-s + 4·106-s − 2·117-s + 121-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3907989100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3907989100\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.902171554130202618911694926166, −8.836511413520278058370010727934, −8.299576845134420884616515967567, −7.933317147152552473405151975889, −7.45507297238138286722127694965, −7.23348364605972382626051290849, −6.92857563484196563808035938517, −6.69149825291659222536667894203, −6.27744887887884551330707218521, −5.86221188537596506521788798599, −5.16527165361072307011411089017, −4.73206168258376016760797597991, −4.56821751285816804460011112535, −3.84131042076333239670533691248, −2.99710921713900790186056301790, −2.94083115754563720068588777951, −2.21260851307020926107644850563, −1.98330077646091093478117271983, −1.36709598565501862331938183107, −0.51974453057723587216119713507,
0.51974453057723587216119713507, 1.36709598565501862331938183107, 1.98330077646091093478117271983, 2.21260851307020926107644850563, 2.94083115754563720068588777951, 2.99710921713900790186056301790, 3.84131042076333239670533691248, 4.56821751285816804460011112535, 4.73206168258376016760797597991, 5.16527165361072307011411089017, 5.86221188537596506521788798599, 6.27744887887884551330707218521, 6.69149825291659222536667894203, 6.92857563484196563808035938517, 7.23348364605972382626051290849, 7.45507297238138286722127694965, 7.933317147152552473405151975889, 8.299576845134420884616515967567, 8.836511413520278058370010727934, 8.902171554130202618911694926166
