| L(s) = 1 | − 2-s − 3-s + 6-s − 4·7-s + 8-s + 3·9-s − 6·11-s + 6·13-s + 4·14-s − 16-s + 2·17-s − 3·18-s + 7·19-s + 4·21-s + 6·22-s − 8·23-s − 24-s − 6·26-s − 8·27-s + 2·29-s − 16·31-s + 6·33-s − 2·34-s − 16·37-s − 7·38-s − 6·39-s − 5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 9-s − 1.80·11-s + 1.66·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s + 1.60·19-s + 0.872·21-s + 1.27·22-s − 1.66·23-s − 0.204·24-s − 1.17·26-s − 1.53·27-s + 0.371·29-s − 2.87·31-s + 1.04·33-s − 0.342·34-s − 2.63·37-s − 1.13·38-s − 0.960·39-s − 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1608622091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1608622091\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 15 T + 128 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−10.36859661640223438742265491932, −9.733230303290900359360727722307, −9.723062748053496413028711491741, −8.925555673642421527702407336663, −8.809603457363235577501433377780, −8.016702852100782720064155637798, −7.80166390321345304139152639750, −7.20960670757774901493112066904, −7.12708564082600825366052882373, −6.31965801127772253482119207790, −6.00537553320877504752555515351, −5.39971952835208150949969959237, −5.35379623129920964278335808955, −4.54638968321674808446313788390, −3.68746108087346518524154159102, −3.47455028564427821478408444633, −3.11755097200733550119288067858, −1.86194946388814138926680489730, −1.55806517875272411449924869470, −0.21899355817458887370701028620,
0.21899355817458887370701028620, 1.55806517875272411449924869470, 1.86194946388814138926680489730, 3.11755097200733550119288067858, 3.47455028564427821478408444633, 3.68746108087346518524154159102, 4.54638968321674808446313788390, 5.35379623129920964278335808955, 5.39971952835208150949969959237, 6.00537553320877504752555515351, 6.31965801127772253482119207790, 7.12708564082600825366052882373, 7.20960670757774901493112066904, 7.80166390321345304139152639750, 8.016702852100782720064155637798, 8.809603457363235577501433377780, 8.925555673642421527702407336663, 9.723062748053496413028711491741, 9.733230303290900359360727722307, 10.36859661640223438742265491932
