| L(s) = 1 | − 2-s − 3-s + 6-s + 5·7-s + 8-s + 3·9-s + 11-s + 10·13-s − 5·14-s − 16-s − 6·17-s − 3·18-s − 2·19-s − 5·21-s − 22-s − 6·23-s − 24-s + 5·25-s − 10·26-s − 8·27-s + 6·29-s − 8·31-s − 33-s + 6·34-s − 2·37-s + 2·38-s − 10·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.408·6-s + 1.88·7-s + 0.353·8-s + 9-s + 0.301·11-s + 2.77·13-s − 1.33·14-s − 1/4·16-s − 1.45·17-s − 0.707·18-s − 0.458·19-s − 1.09·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s + 25-s − 1.96·26-s − 1.53·27-s + 1.11·29-s − 1.43·31-s − 0.174·33-s + 1.02·34-s − 0.328·37-s + 0.324·38-s − 1.60·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9074482733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9074482733\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−13.18296850553675620150020947792, −12.85862869679776401122756751188, −11.76035927782490527261700097193, −11.71995485357379287464386892907, −11.08305274011583363056940299549, −10.76062740832841701531708634085, −10.40936079526681555390015911412, −9.724108494438195848225714440184, −8.675946166386378985666300410923, −8.672879547625767873225993447290, −8.321284399952509837047196210056, −7.56057471469725303771995895587, −6.69803946855028792913509123991, −6.50983476260970354200760039868, −5.51121264885572855135431695107, −5.00308750393004902050806386034, −4.05371895049646775174951995858, −3.90394753737404497208525797759, −1.88776775721159133254667905240, −1.38688820824962980499379045928,
1.38688820824962980499379045928, 1.88776775721159133254667905240, 3.90394753737404497208525797759, 4.05371895049646775174951995858, 5.00308750393004902050806386034, 5.51121264885572855135431695107, 6.50983476260970354200760039868, 6.69803946855028792913509123991, 7.56057471469725303771995895587, 8.321284399952509837047196210056, 8.672879547625767873225993447290, 8.675946166386378985666300410923, 9.724108494438195848225714440184, 10.40936079526681555390015911412, 10.76062740832841701531708634085, 11.08305274011583363056940299549, 11.71995485357379287464386892907, 11.76035927782490527261700097193, 12.85862869679776401122756751188, 13.18296850553675620150020947792
