| L(s) = 1 | + 3·4-s + 6·9-s − 12·11-s + 5·16-s + 4·19-s + 2·29-s + 4·31-s + 18·36-s + 4·41-s − 36·44-s + 10·49-s + 16·59-s − 12·61-s + 3·64-s − 24·71-s + 12·76-s + 20·79-s + 27·81-s − 36·89-s − 72·99-s + 20·101-s + 28·109-s + 6·116-s + 86·121-s + 12·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 2·9-s − 3.61·11-s + 5/4·16-s + 0.917·19-s + 0.371·29-s + 0.718·31-s + 3·36-s + 0.624·41-s − 5.42·44-s + 10/7·49-s + 2.08·59-s − 1.53·61-s + 3/8·64-s − 2.84·71-s + 1.37·76-s + 2.25·79-s + 3·81-s − 3.81·89-s − 7.23·99-s + 1.99·101-s + 2.68·109-s + 0.557·116-s + 7.81·121-s + 1.07·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.804374662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.804374662\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.67312196610127919462588598658, −10.34072067206267059604203245946, −9.833167495559740938355149837873, −9.776237915665878080063535621343, −8.790289877698730065584705724601, −8.276166332733417785109124347038, −7.77598778692835818643125792560, −7.40411307127876632890816351909, −7.37827757031045747278894894102, −6.87980060839074380493989575229, −6.22198069079057810508297735477, −5.51051609302843871070980223670, −5.49969097173598607800637660478, −4.56063402080358753068589588525, −4.46914927729868417741466153421, −3.29871767503832252663994663052, −2.92347578162155466803677423313, −2.34101009603683483821703918774, −1.91263034095113813180296094491, −0.862835805039292746039049720786,
0.862835805039292746039049720786, 1.91263034095113813180296094491, 2.34101009603683483821703918774, 2.92347578162155466803677423313, 3.29871767503832252663994663052, 4.46914927729868417741466153421, 4.56063402080358753068589588525, 5.49969097173598607800637660478, 5.51051609302843871070980223670, 6.22198069079057810508297735477, 6.87980060839074380493989575229, 7.37827757031045747278894894102, 7.40411307127876632890816351909, 7.77598778692835818643125792560, 8.276166332733417785109124347038, 8.790289877698730065584705724601, 9.776237915665878080063535621343, 9.833167495559740938355149837873, 10.34072067206267059604203245946, 10.67312196610127919462588598658
