| L(s) = 1 | + 3-s − 3·5-s + 3·11-s − 3·15-s − 6·17-s − 2·19-s + 25-s − 27-s + 18·29-s + 5·31-s + 3·33-s − 10·37-s + 12·47-s − 6·51-s + 9·53-s − 9·55-s − 2·57-s − 9·59-s − 24·67-s + 12·73-s + 75-s + 9·79-s − 81-s − 6·83-s + 18·85-s + 18·87-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 0.904·11-s − 0.774·15-s − 1.45·17-s − 0.458·19-s + 1/5·25-s − 0.192·27-s + 3.34·29-s + 0.898·31-s + 0.522·33-s − 1.64·37-s + 1.75·47-s − 0.840·51-s + 1.23·53-s − 1.21·55-s − 0.264·57-s − 1.17·59-s − 2.93·67-s + 1.40·73-s + 0.115·75-s + 1.01·79-s − 1/9·81-s − 0.658·83-s + 1.95·85-s + 1.92·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.732219368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.732219368\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 24 T + 259 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T + 101 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−8.952111334548776175456974282711, −8.855426110933115167239576857527, −8.472143172246000693225470748079, −7.944839739836297320405585783842, −7.88454516855995810949953044389, −7.24605705839342267284631427107, −6.78901410403490639059555040879, −6.68485477892148032298159891672, −6.17263184419040992947263615415, −5.79074193678024563218607245976, −4.95070527027467724661806520015, −4.62522903507757384733851379372, −4.38124431498908982707228474431, −3.88160556942459854174710190481, −3.57790410341216103499082643008, −2.89028599935860106670883267044, −2.60934635459595210091445547858, −1.96589583707472775622036054929, −1.19258533009139570742679586053, −0.47187175054487656799691651162,
0.47187175054487656799691651162, 1.19258533009139570742679586053, 1.96589583707472775622036054929, 2.60934635459595210091445547858, 2.89028599935860106670883267044, 3.57790410341216103499082643008, 3.88160556942459854174710190481, 4.38124431498908982707228474431, 4.62522903507757384733851379372, 4.95070527027467724661806520015, 5.79074193678024563218607245976, 6.17263184419040992947263615415, 6.68485477892148032298159891672, 6.78901410403490639059555040879, 7.24605705839342267284631427107, 7.88454516855995810949953044389, 7.944839739836297320405585783842, 8.472143172246000693225470748079, 8.855426110933115167239576857527, 8.952111334548776175456974282711
