| L(s) = 1 | − 3-s + 7-s − 13-s − 2·19-s − 21-s + 2·25-s + 27-s − 2·31-s − 2·37-s + 39-s + 43-s + 49-s + 2·57-s + 61-s + 67-s − 2·73-s − 2·75-s − 2·79-s − 81-s − 91-s + 2·93-s + 97-s − 2·103-s − 2·109-s + 2·111-s − 121-s + 127-s + ⋯ |
| L(s) = 1 | − 3-s + 7-s − 13-s − 2·19-s − 21-s + 2·25-s + 27-s − 2·31-s − 2·37-s + 39-s + 43-s + 49-s + 2·57-s + 61-s + 67-s − 2·73-s − 2·75-s − 2·79-s − 81-s − 91-s + 2·93-s + 97-s − 2·103-s − 2·109-s + 2·111-s − 121-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3304947556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3304947556\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−13.13712256832366131695424890221, −12.87467162212949865026756229537, −12.19828706702413409649508990299, −12.16223898308493711616487105926, −11.21618870351214128473596623863, −11.07231596751651044830586872506, −10.42865324778073088288378406500, −10.34906402372767019976987561544, −9.223432634792648971063271686512, −8.761415698778100303896616633042, −8.426320178916794271269645742399, −7.61222166988859049414115298574, −6.82635072436709119164717233769, −6.78205128267635501739595218871, −5.53519951937327730915506033297, −5.45461149728077424722847653941, −4.64419865390129387614011547920, −4.12196296708099721647063286021, −2.86001860257039679398210868800, −1.85492671042857562741002517517,
1.85492671042857562741002517517, 2.86001860257039679398210868800, 4.12196296708099721647063286021, 4.64419865390129387614011547920, 5.45461149728077424722847653941, 5.53519951937327730915506033297, 6.78205128267635501739595218871, 6.82635072436709119164717233769, 7.61222166988859049414115298574, 8.426320178916794271269645742399, 8.761415698778100303896616633042, 9.223432634792648971063271686512, 10.34906402372767019976987561544, 10.42865324778073088288378406500, 11.07231596751651044830586872506, 11.21618870351214128473596623863, 12.16223898308493711616487105926, 12.19828706702413409649508990299, 12.87467162212949865026756229537, 13.13712256832366131695424890221
