| L(s) = 1 | − 2-s + 3-s + 2·4-s + 5-s − 6-s − 5·8-s − 10-s + 2·12-s + 12·13-s + 15-s + 5·16-s + 2·17-s − 8·19-s + 2·20-s − 8·23-s − 5·24-s − 12·26-s − 27-s − 4·29-s − 30-s + 4·31-s − 10·32-s − 2·34-s + 2·37-s + 8·38-s + 12·39-s − 5·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.408·6-s − 1.76·8-s − 0.316·10-s + 0.577·12-s + 3.32·13-s + 0.258·15-s + 5/4·16-s + 0.485·17-s − 1.83·19-s + 0.447·20-s − 1.66·23-s − 1.02·24-s − 2.35·26-s − 0.192·27-s − 0.742·29-s − 0.182·30-s + 0.718·31-s − 1.76·32-s − 0.342·34-s + 0.328·37-s + 1.29·38-s + 1.92·39-s − 0.790·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.066049686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066049686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−10.56324952502121165516736973958, −10.21513019721889744588714716247, −9.673084331152652731725376117625, −9.142671875229193952026188171827, −8.714492154223897861050509021309, −8.652693295096881803388888080875, −8.287735619010345288068790827385, −7.46699723572523875021634620936, −7.45495127667200245001312433880, −6.30232677462348876300017408378, −6.10687461110529749998803352213, −6.06215823738007843248542834061, −5.75599649755096734032464695844, −4.38811972194348669394918723538, −4.05053938266912190787151039746, −3.43584376871147869793226047999, −2.99376094750741457280557848510, −2.17265511341079055014304820807, −1.79144008333543833352681486237, −0.816880209091725251028262654467,
0.816880209091725251028262654467, 1.79144008333543833352681486237, 2.17265511341079055014304820807, 2.99376094750741457280557848510, 3.43584376871147869793226047999, 4.05053938266912190787151039746, 4.38811972194348669394918723538, 5.75599649755096734032464695844, 6.06215823738007843248542834061, 6.10687461110529749998803352213, 6.30232677462348876300017408378, 7.45495127667200245001312433880, 7.46699723572523875021634620936, 8.287735619010345288068790827385, 8.652693295096881803388888080875, 8.714492154223897861050509021309, 9.142671875229193952026188171827, 9.673084331152652731725376117625, 10.21513019721889744588714716247, 10.56324952502121165516736973958
