| L(s) = 1 | + 3-s + 3·5-s − 7-s − 3·11-s + 3·15-s + 6·17-s − 2·19-s − 21-s + 25-s − 27-s + 18·29-s + 5·31-s − 3·33-s − 3·35-s − 10·37-s + 12·47-s − 6·49-s + 6·51-s + 9·53-s − 9·55-s − 2·57-s − 9·59-s + 24·67-s − 12·73-s + 75-s + 3·77-s − 9·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 0.377·7-s − 0.904·11-s + 0.774·15-s + 1.45·17-s − 0.458·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 3.34·29-s + 0.898·31-s − 0.522·33-s − 0.507·35-s − 1.64·37-s + 1.75·47-s − 6/7·49-s + 0.840·51-s + 1.23·53-s − 1.21·55-s − 0.264·57-s − 1.17·59-s + 2.93·67-s − 1.40·73-s + 0.115·75-s + 0.341·77-s − 1.01·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.309625824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.309625824\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 24 T + 259 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T + 101 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−11.89283928389605238915491261049, −11.33807669996562159470952058435, −10.39770519244131540883513627285, −10.32607441806919193274367833830, −10.09236921106327671604554299848, −9.594306911347874855998039066891, −8.975605550546403207334380693422, −8.458415058171009205841618221691, −8.200417466545870987154485219810, −7.60754302499055344656968711218, −6.83417513101135604906306015068, −6.56596142331219444038537919859, −5.77456021823975596808470790083, −5.55437970291621167158891736823, −4.87693053333508069501427701415, −4.20967410185630123000259716896, −3.25300767388092965948078299974, −2.78302971185410998715134721106, −2.22489091886172336378924877710, −1.15085164351011270954554162308,
1.15085164351011270954554162308, 2.22489091886172336378924877710, 2.78302971185410998715134721106, 3.25300767388092965948078299974, 4.20967410185630123000259716896, 4.87693053333508069501427701415, 5.55437970291621167158891736823, 5.77456021823975596808470790083, 6.56596142331219444038537919859, 6.83417513101135604906306015068, 7.60754302499055344656968711218, 8.200417466545870987154485219810, 8.458415058171009205841618221691, 8.975605550546403207334380693422, 9.594306911347874855998039066891, 10.09236921106327671604554299848, 10.32607441806919193274367833830, 10.39770519244131540883513627285, 11.33807669996562159470952058435, 11.89283928389605238915491261049
