| L(s) = 1 | + 5-s + 9-s + 19-s + 29-s + 2·41-s + 45-s − 2·49-s − 3·59-s − 61-s + 3·71-s − 3·79-s + 89-s + 95-s + 101-s − 109-s − 121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 5-s + 9-s + 19-s + 29-s + 2·41-s + 45-s − 2·49-s − 3·59-s − 61-s + 3·71-s − 3·79-s + 89-s + 95-s + 101-s − 109-s − 121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.568561264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.568561264\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.735047761983297755658855819874, −9.603719215726221964601517760798, −9.054368325783819712787400774422, −8.964108096063200209655686556520, −8.051979499326616417730632641467, −7.917319741949913610651196121430, −7.50009745542313518702915859435, −7.06007525636684195699814796283, −6.42957784686699256988661388612, −6.34914479170518942619213079663, −5.79716182483802234035147837302, −5.38045434340819595387331934285, −4.75491875274002610518996741225, −4.59157677232805852620433497212, −3.97860389539106826538131796932, −3.33741877174442598533052928101, −2.87179984536598896766796647761, −2.30532721374326435567173393619, −1.56852212430036046910997973783, −1.19479593689736668105221996145,
1.19479593689736668105221996145, 1.56852212430036046910997973783, 2.30532721374326435567173393619, 2.87179984536598896766796647761, 3.33741877174442598533052928101, 3.97860389539106826538131796932, 4.59157677232805852620433497212, 4.75491875274002610518996741225, 5.38045434340819595387331934285, 5.79716182483802234035147837302, 6.34914479170518942619213079663, 6.42957784686699256988661388612, 7.06007525636684195699814796283, 7.50009745542313518702915859435, 7.917319741949913610651196121430, 8.051979499326616417730632641467, 8.964108096063200209655686556520, 9.054368325783819712787400774422, 9.603719215726221964601517760798, 9.735047761983297755658855819874
