| L(s) = 1 | − 2-s + 16·4-s − 47·8-s − 81·9-s + 206·11-s + 47·16-s + 81·18-s − 206·22-s + 734·23-s − 625·25-s + 2.46e3·29-s − 752·32-s − 1.29e3·36-s + 1.29e3·37-s − 668·43-s + 3.29e3·44-s − 734·46-s + 625·50-s + 5.58e3·53-s − 2.46e3·58-s − 1.88e3·64-s − 4.94e3·67-s + 5.82e3·71-s + 3.80e3·72-s − 1.29e3·74-s + 3.64e3·79-s + 668·86-s + ⋯ |
| L(s) = 1 | − 1/4·2-s + 4-s − 0.734·8-s − 9-s + 1.70·11-s + 0.183·16-s + 1/4·18-s − 0.425·22-s + 1.38·23-s − 25-s + 2.93·29-s − 0.734·32-s − 36-s + 0.945·37-s − 0.361·43-s + 1.70·44-s − 0.346·46-s + 1/4·50-s + 1.98·53-s − 0.733·58-s − 0.460·64-s − 1.10·67-s + 1.15·71-s + 0.734·72-s − 0.236·74-s + 0.584·79-s + 0.0903·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.106013261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.106013261\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - 15 T^{2} + p^{4} T^{3} + p^{8} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 206 T + 27795 T^{2} - 206 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 734 T + 258915 T^{2} - 734 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 1234 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 1294 T - 199725 T^{2} - 1294 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 334 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 5582 T + 23268243 T^{2} - 5582 p^{4} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4946 T + 4311795 T^{2} + 4946 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2914 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 3646 T - 25656765 T^{2} - 3646 p^{4} T^{3} + p^{8} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
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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
−15.24961562831167636606685899898, −14.60144560973149915620499630153, −14.10440286114921863442859588217, −13.60382604080703148038476002507, −12.64505158838501834716532760603, −11.82661709222005691312348018295, −11.80032090769131054027101039065, −11.21822405260010153483571339005, −10.47007656057678936234828793449, −9.737257815540610610359955395362, −8.887337718285025636834792151351, −8.676759312273618569918526911138, −7.67744387118201432369994400349, −6.67524590754859883805106535306, −6.50657966749115155852874810212, −5.63657384310397046760815961931, −4.45863280702003633656854553125, −3.27881667833585164453089653318, −2.43063663150742546720321985454, −0.987027341325274920568579000502,
0.987027341325274920568579000502, 2.43063663150742546720321985454, 3.27881667833585164453089653318, 4.45863280702003633656854553125, 5.63657384310397046760815961931, 6.50657966749115155852874810212, 6.67524590754859883805106535306, 7.67744387118201432369994400349, 8.676759312273618569918526911138, 8.887337718285025636834792151351, 9.737257815540610610359955395362, 10.47007656057678936234828793449, 11.21822405260010153483571339005, 11.80032090769131054027101039065, 11.82661709222005691312348018295, 12.64505158838501834716532760603, 13.60382604080703148038476002507, 14.10440286114921863442859588217, 14.60144560973149915620499630153, 15.24961562831167636606685899898
