| L(s) = 1 | + 5-s + 2·11-s − 6·17-s − 2·19-s + 3·23-s + 4·29-s + 5·31-s + 20·37-s + 6·41-s + 6·43-s + 8·47-s + 7·49-s − 6·53-s + 2·55-s − 5·61-s + 2·67-s − 4·71-s + 12·73-s + 11·79-s + 9·83-s − 6·85-s − 20·89-s − 2·95-s − 8·97-s − 12·101-s + 12·103-s + 8·107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.603·11-s − 1.45·17-s − 0.458·19-s + 0.625·23-s + 0.742·29-s + 0.898·31-s + 3.28·37-s + 0.937·41-s + 0.914·43-s + 1.16·47-s + 49-s − 0.824·53-s + 0.269·55-s − 0.640·61-s + 0.244·67-s − 0.474·71-s + 1.40·73-s + 1.23·79-s + 0.987·83-s − 0.650·85-s − 2.11·89-s − 0.205·95-s − 0.812·97-s − 1.19·101-s + 1.18·103-s + 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.678981155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.678981155\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.771845772725393772005347264934, −8.633640529368152430870388309197, −7.941890736681617848041627856411, −7.935129861964059786598461546339, −7.18436131375254493863215124108, −7.06842397019783576798061920777, −6.37144729626800161346533947602, −6.35634207652174261122049187729, −5.86094092563580165313679689839, −5.62104538425716082708062795148, −4.80259045979508856298416584249, −4.58796546941800055053733911465, −4.18099065835868493197725825981, −4.01439755456307918209020040734, −3.03631727904858194101487133208, −2.86391029664859266550288372116, −2.20541625893556350196532544586, −2.01146102515500321100595290234, −0.904509761382959684226773690652, −0.78389592068946060390318130506,
0.78389592068946060390318130506, 0.904509761382959684226773690652, 2.01146102515500321100595290234, 2.20541625893556350196532544586, 2.86391029664859266550288372116, 3.03631727904858194101487133208, 4.01439755456307918209020040734, 4.18099065835868493197725825981, 4.58796546941800055053733911465, 4.80259045979508856298416584249, 5.62104538425716082708062795148, 5.86094092563580165313679689839, 6.35634207652174261122049187729, 6.37144729626800161346533947602, 7.06842397019783576798061920777, 7.18436131375254493863215124108, 7.935129861964059786598461546339, 7.941890736681617848041627856411, 8.633640529368152430870388309197, 8.771845772725393772005347264934
