| L(s) = 1 | − 2·3-s + 2·7-s + 2·9-s + 8·13-s + 8·17-s − 8·19-s − 4·21-s + 10·23-s − 6·27-s − 16·39-s − 8·41-s − 14·43-s + 6·47-s + 2·49-s − 16·51-s + 8·53-s + 16·57-s − 8·59-s + 16·61-s + 4·63-s + 6·67-s − 20·69-s − 8·73-s − 16·79-s + 11·81-s + 10·83-s + 16·91-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 2/3·9-s + 2.21·13-s + 1.94·17-s − 1.83·19-s − 0.872·21-s + 2.08·23-s − 1.15·27-s − 2.56·39-s − 1.24·41-s − 2.13·43-s + 0.875·47-s + 2/7·49-s − 2.24·51-s + 1.09·53-s + 2.11·57-s − 1.04·59-s + 2.04·61-s + 0.503·63-s + 0.733·67-s − 2.40·69-s − 0.936·73-s − 1.80·79-s + 11/9·81-s + 1.09·83-s + 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.835157122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.835157122\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−9.799213305508140722053793563820, −8.933674618950114818973548524953, −8.892890613355382179745385989393, −8.288093545460029554950883936100, −8.244102082229293383318882509651, −7.58955111222279669233963023116, −7.11515040985469529459204446923, −6.57925370113038588392895013871, −6.49187949885099807548182026998, −5.85807656234190864984834218509, −5.37907733006028821695365489587, −5.35600697270186469901385472790, −4.76900576626099330319206888718, −4.04886027243127850007050633577, −3.82921008566749396781685750606, −3.28228076936687235527126775812, −2.63970096018183434443584677962, −1.52306688034955979937522293970, −1.48924525087183715637078100899, −0.62363932292561094271429362883,
0.62363932292561094271429362883, 1.48924525087183715637078100899, 1.52306688034955979937522293970, 2.63970096018183434443584677962, 3.28228076936687235527126775812, 3.82921008566749396781685750606, 4.04886027243127850007050633577, 4.76900576626099330319206888718, 5.35600697270186469901385472790, 5.37907733006028821695365489587, 5.85807656234190864984834218509, 6.49187949885099807548182026998, 6.57925370113038588392895013871, 7.11515040985469529459204446923, 7.58955111222279669233963023116, 8.244102082229293383318882509651, 8.288093545460029554950883936100, 8.892890613355382179745385989393, 8.933674618950114818973548524953, 9.799213305508140722053793563820
