| L(s) = 1 | + 2·3-s + 9-s + 16·23-s + 6·25-s − 4·27-s − 12·37-s − 8·47-s + 49-s + 12·59-s + 8·61-s + 32·69-s − 28·73-s + 12·75-s − 11·81-s + 12·83-s − 4·97-s − 24·107-s + 20·109-s − 24·111-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 16·141-s + 2·147-s + 149-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 3.33·23-s + 6/5·25-s − 0.769·27-s − 1.97·37-s − 1.16·47-s + 1/7·49-s + 1.56·59-s + 1.02·61-s + 3.85·69-s − 3.27·73-s + 1.38·75-s − 1.22·81-s + 1.31·83-s − 0.406·97-s − 2.32·107-s + 1.91·109-s − 2.27·111-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.34·141-s + 0.164·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.075551686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.075551686\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.905679050168767306734287948816, −9.277412843087557691575843134492, −8.928329697232844766585907221767, −8.577356052281653482347188544077, −8.188057214509100026281306554311, −7.19524746076483701675661172135, −7.14438434722230698367292292955, −6.56786833309649857729048855391, −5.54310497317186675887471826542, −5.10863365890477760326382783699, −4.50185552371175935345657331179, −3.47782444964075224023522277637, −3.14137792992390816749783057891, −2.45324653241146232194564495570, −1.30261481912190858285121945099,
1.30261481912190858285121945099, 2.45324653241146232194564495570, 3.14137792992390816749783057891, 3.47782444964075224023522277637, 4.50185552371175935345657331179, 5.10863365890477760326382783699, 5.54310497317186675887471826542, 6.56786833309649857729048855391, 7.14438434722230698367292292955, 7.19524746076483701675661172135, 8.188057214509100026281306554311, 8.577356052281653482347188544077, 8.928329697232844766585907221767, 9.277412843087557691575843134492, 9.905679050168767306734287948816
