| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 3·8-s − 9-s − 4·11-s − 12-s + 13-s − 2·14-s + 16-s + 2·17-s + 18-s − 2·21-s + 4·22-s − 3·23-s + 3·24-s − 6·25-s − 26-s + 2·28-s + 8·29-s + 32-s + 4·33-s − 2·34-s − 36-s + 10·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 1.06·8-s − 1/3·9-s − 1.20·11-s − 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.436·21-s + 0.852·22-s − 0.625·23-s + 0.612·24-s − 6/5·25-s − 0.196·26-s + 0.377·28-s + 1.48·29-s + 0.176·32-s + 0.696·33-s − 0.342·34-s − 1/6·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 997 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 997 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3373378289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3373378289\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 997 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 26 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 66 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 19 T + 190 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T - 10 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 132 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−19.9579900306, −19.4486549550, −18.6163968218, −18.2072823251, −17.8711919846, −17.3574667781, −16.7921087267, −15.8471617323, −15.8082953901, −15.0139797296, −14.2640324879, −13.6819735905, −12.8251906707, −12.0176762276, −11.6503683921, −10.9555534550, −10.3755933677, −9.60369407405, −8.79679767998, −7.98628403176, −7.54575111696, −6.13928265663, −5.75896104005, −4.51194252969, −2.68687741657,
2.68687741657, 4.51194252969, 5.75896104005, 6.13928265663, 7.54575111696, 7.98628403176, 8.79679767998, 9.60369407405, 10.3755933677, 10.9555534550, 11.6503683921, 12.0176762276, 12.8251906707, 13.6819735905, 14.2640324879, 15.0139797296, 15.8082953901, 15.8471617323, 16.7921087267, 17.3574667781, 17.8711919846, 18.2072823251, 18.6163968218, 19.4486549550, 19.9579900306
