| L(s) = 1 | + 2·2-s − 3-s + 9·5-s − 2·6-s + 2·7-s − 8·8-s + 9·9-s + 18·10-s − 17·11-s + 4·14-s − 9·15-s − 16·16-s + 25·17-s + 18·18-s + 7·19-s − 2·21-s − 34·22-s − 9·23-s + 8·24-s + 29·25-s − 26·27-s − 18·30-s − 57·31-s + 17·33-s + 50·34-s + 18·35-s − 15·37-s + ⋯ |
| L(s) = 1 | + 2-s − 1/3·3-s + 9/5·5-s − 1/3·6-s + 2/7·7-s − 8-s + 9-s + 9/5·10-s − 1.54·11-s + 2/7·14-s − 3/5·15-s − 16-s + 1.47·17-s + 18-s + 7/19·19-s − 0.0952·21-s − 1.54·22-s − 0.391·23-s + 1/3·24-s + 1.15·25-s − 0.962·27-s − 3/5·30-s − 1.83·31-s + 0.515·33-s + 1.47·34-s + 0.514·35-s − 0.405·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.142428870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.142428870\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - 2 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 8 T^{2} + p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T + 168 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T + 336 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7 T - 312 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 556 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1490 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 57 T + 2044 T^{2} + 57 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 15 T + 1444 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 87 T + 4732 T^{2} - 87 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p T + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 55 T - 456 T^{2} - 55 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 39 T + 4228 T^{2} + 39 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 17 T - 4200 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 119 T + 8832 T^{2} + 119 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 129 T + 11788 T^{2} + 129 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 71 T - 2880 T^{2} + 71 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
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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
−15.17403622682071436135104518944, −14.33577214826405995789610168542, −14.25620008177459148997664316242, −13.56733413238691319101463175455, −13.07037479468377140867206428889, −12.69272922329765142296715464780, −12.31611519321041910908629344339, −11.31494944788452343117595600659, −10.62386795813253991585747455639, −10.04861619722270340940544451561, −9.586981328631099635582231613705, −9.067733118725804416564230730455, −7.83408144860892100665282894309, −7.36371868579076417158386296459, −6.13373592708398675204763505983, −5.62715576423065889018193210049, −5.32654353467139726133363162300, −4.37509473352939048165451832031, −3.13308585957546575413998737888, −1.89530875372495892808619825282,
1.89530875372495892808619825282, 3.13308585957546575413998737888, 4.37509473352939048165451832031, 5.32654353467139726133363162300, 5.62715576423065889018193210049, 6.13373592708398675204763505983, 7.36371868579076417158386296459, 7.83408144860892100665282894309, 9.067733118725804416564230730455, 9.586981328631099635582231613705, 10.04861619722270340940544451561, 10.62386795813253991585747455639, 11.31494944788452343117595600659, 12.31611519321041910908629344339, 12.69272922329765142296715464780, 13.07037479468377140867206428889, 13.56733413238691319101463175455, 14.25620008177459148997664316242, 14.33577214826405995789610168542, 15.17403622682071436135104518944
