| L(s) = 1 | + 4·5-s + 6·7-s + 8·11-s − 5·13-s + 8·19-s − 4·23-s + 5·25-s + 8·29-s − 2·31-s + 24·35-s − 10·37-s + 8·41-s − 5·43-s + 8·47-s + 13·49-s + 4·53-s + 32·55-s − 12·59-s + 61-s − 20·65-s + 3·67-s − 16·71-s + 15·73-s + 48·77-s − 7·79-s − 12·89-s − 30·91-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2.26·7-s + 2.41·11-s − 1.38·13-s + 1.83·19-s − 0.834·23-s + 25-s + 1.48·29-s − 0.359·31-s + 4.05·35-s − 1.64·37-s + 1.24·41-s − 0.762·43-s + 1.16·47-s + 13/7·49-s + 0.549·53-s + 4.31·55-s − 1.56·59-s + 0.128·61-s − 2.48·65-s + 0.366·67-s − 1.89·71-s + 1.75·73-s + 5.47·77-s − 0.787·79-s − 1.27·89-s − 3.14·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.095042780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.095042780\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 16 T + 185 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 15 T + 152 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.241431494376915632436297351459, −8.843196246827819270476044869076, −8.131730964228801101711620765473, −8.029850932224612115333896879655, −7.32817658501532653875286554678, −7.26598114923559937606200382408, −6.73556085055988780283667544223, −6.13993357742049147799532149302, −6.08866185893380106237319383682, −5.34167253995475635532829145136, −5.19549848884685461009015931338, −4.90668742092623487607894746040, −4.20094931029252274647204638510, −4.13608802513676338527472232413, −3.31222740790048070237403236388, −2.72696892334717888625116720639, −2.15422673641753045623852449635, −1.60887496822223174216876199303, −1.54929220257768350982780903308, −0.932705803029573658256122849041,
0.932705803029573658256122849041, 1.54929220257768350982780903308, 1.60887496822223174216876199303, 2.15422673641753045623852449635, 2.72696892334717888625116720639, 3.31222740790048070237403236388, 4.13608802513676338527472232413, 4.20094931029252274647204638510, 4.90668742092623487607894746040, 5.19549848884685461009015931338, 5.34167253995475635532829145136, 6.08866185893380106237319383682, 6.13993357742049147799532149302, 6.73556085055988780283667544223, 7.26598114923559937606200382408, 7.32817658501532653875286554678, 8.029850932224612115333896879655, 8.131730964228801101711620765473, 8.843196246827819270476044869076, 9.241431494376915632436297351459
