| L(s) = 1 | − 2·2-s + 3-s + 3·4-s + 5-s − 2·6-s − 3·7-s − 4·8-s − 2·10-s + 11-s + 3·12-s + 6·14-s + 15-s + 5·16-s − 2·17-s − 2·19-s + 3·20-s − 3·21-s − 2·22-s − 8·23-s − 4·24-s − 27-s − 9·28-s − 14·29-s − 2·30-s + 7·31-s − 6·32-s + 33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.447·5-s − 0.816·6-s − 1.13·7-s − 1.41·8-s − 0.632·10-s + 0.301·11-s + 0.866·12-s + 1.60·14-s + 0.258·15-s + 5/4·16-s − 0.485·17-s − 0.458·19-s + 0.670·20-s − 0.654·21-s − 0.426·22-s − 1.66·23-s − 0.816·24-s − 0.192·27-s − 1.70·28-s − 2.59·29-s − 0.365·30-s + 1.25·31-s − 1.06·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5819813553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5819813553\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 31 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−9.990708414421564382708284190724, −9.752915314096384660635689240277, −9.554737234997355804079623133824, −9.142657534957722120597509163953, −8.557957725351166902086562166440, −8.282725662691111679503335049537, −8.015751975153089056442500790047, −7.40800210902523445082908965929, −6.83945226507378624630101112173, −6.56099427724685621792942273860, −6.33257837740565716810340677601, −5.52310704239098164005590905895, −5.38111751583812566778080204148, −4.29530707641256811094705599474, −3.70186392474151333235598383124, −3.42660656730721717620560887502, −2.61755259340189275410882636014, −2.07000083139532324582035261716, −1.70649617039221832800780428782, −0.41405623119766706184646915098,
0.41405623119766706184646915098, 1.70649617039221832800780428782, 2.07000083139532324582035261716, 2.61755259340189275410882636014, 3.42660656730721717620560887502, 3.70186392474151333235598383124, 4.29530707641256811094705599474, 5.38111751583812566778080204148, 5.52310704239098164005590905895, 6.33257837740565716810340677601, 6.56099427724685621792942273860, 6.83945226507378624630101112173, 7.40800210902523445082908965929, 8.015751975153089056442500790047, 8.282725662691111679503335049537, 8.557957725351166902086562166440, 9.142657534957722120597509163953, 9.554737234997355804079623133824, 9.752915314096384660635689240277, 9.990708414421564382708284190724
