| L(s) = 1 | + 2-s + 3-s + 6-s − 8-s − 3·11-s − 5·13-s − 16-s + 6·17-s + 6·19-s − 3·22-s − 5·23-s − 24-s − 5·26-s − 27-s − 2·29-s − 12·31-s − 3·33-s + 6·34-s − 9·37-s + 6·38-s − 5·39-s + 10·41-s − 8·43-s − 5·46-s + 24·47-s − 48-s + 7·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.408·6-s − 0.353·8-s − 0.904·11-s − 1.38·13-s − 1/4·16-s + 1.45·17-s + 1.37·19-s − 0.639·22-s − 1.04·23-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 0.371·29-s − 2.15·31-s − 0.522·33-s + 1.02·34-s − 1.47·37-s + 0.973·38-s − 0.800·39-s + 1.56·41-s − 1.21·43-s − 0.737·46-s + 3.50·47-s − 0.144·48-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.309969802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.309969802\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5 T + 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 7 T - 22 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−9.379284445121763749363212132234, −9.269433627954626966771122673526, −8.506894261582760813379587010005, −8.106203497465271703235156437085, −7.66976962951129295133175342630, −7.58649846770502609005742745322, −7.07377187697153556581486378536, −6.81937349730380775889463938295, −5.84752870125213846692346144179, −5.67056918983759923384111769113, −5.30090831581122833637808914326, −5.19445062846193611484008374396, −4.34280612593254295441653514660, −4.07072308666172391878955236239, −3.43137964025204721084664252944, −3.17746593823139008568495555300, −2.63138453082759171259682994691, −2.14638099502382911961866887984, −1.52088048516929079537687616167, −0.45689508505433751061279314688,
0.45689508505433751061279314688, 1.52088048516929079537687616167, 2.14638099502382911961866887984, 2.63138453082759171259682994691, 3.17746593823139008568495555300, 3.43137964025204721084664252944, 4.07072308666172391878955236239, 4.34280612593254295441653514660, 5.19445062846193611484008374396, 5.30090831581122833637808914326, 5.67056918983759923384111769113, 5.84752870125213846692346144179, 6.81937349730380775889463938295, 7.07377187697153556581486378536, 7.58649846770502609005742745322, 7.66976962951129295133175342630, 8.106203497465271703235156437085, 8.506894261582760813379587010005, 9.269433627954626966771122673526, 9.379284445121763749363212132234
