| L(s) = 1 | − 5-s + 9-s + 4·29-s + 4·41-s − 45-s − 2·61-s + 2·89-s + 2·101-s + 2·109-s − 121-s + 125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 5-s + 9-s + 4·29-s + 4·41-s − 45-s − 2·61-s + 2·89-s + 2·101-s + 2·109-s − 121-s + 125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.545164227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.545164227\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$ | \( ( 1 - T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.702492116265349537293042649560, −8.476584581521341309333890046944, −7.955275465023248645659869494878, −7.77915737582202487182340114251, −7.32983875188193982980792942525, −7.24802992507568953803685822014, −6.44553819269419844744212034115, −6.42174566478572999440638066040, −6.00618644976148158386440807334, −5.51556473355296007245095231125, −4.71309788299406934760416163815, −4.68396227160948575842711348784, −4.31122266114823230233121323007, −4.06327471098678505270898339461, −3.21928571733310263959036936247, −3.17364617860842158216506235416, −2.45327454368982615807571498187, −2.08894897484033772793513713653, −0.990578571973092936812892467527, −0.976050005442044870313045094085,
0.976050005442044870313045094085, 0.990578571973092936812892467527, 2.08894897484033772793513713653, 2.45327454368982615807571498187, 3.17364617860842158216506235416, 3.21928571733310263959036936247, 4.06327471098678505270898339461, 4.31122266114823230233121323007, 4.68396227160948575842711348784, 4.71309788299406934760416163815, 5.51556473355296007245095231125, 6.00618644976148158386440807334, 6.42174566478572999440638066040, 6.44553819269419844744212034115, 7.24802992507568953803685822014, 7.32983875188193982980792942525, 7.77915737582202487182340114251, 7.955275465023248645659869494878, 8.476584581521341309333890046944, 8.702492116265349537293042649560
