| L(s) = 1 | + (1.17 − 1.69i)2-s + (0.792 − 0.0601i)3-s + (−0.788 − 2.09i)4-s + (−0.616 + 1.39i)5-s + (0.827 − 1.41i)6-s + (−4.03 − 2.26i)7-s + (−0.481 − 0.120i)8-s + (−2.34 + 0.357i)9-s + (1.63 + 2.67i)10-s + (1.69 − 0.635i)11-s + (−0.751 − 1.61i)12-s + (−2.26 − 4.04i)13-s + (−8.56 + 4.18i)14-s + (−0.404 + 1.14i)15-s + (2.60 − 2.27i)16-s + (−0.456 − 0.852i)17-s + ⋯ |
| L(s) = 1 | + (0.828 − 1.19i)2-s + (0.457 − 0.0347i)3-s + (−0.394 − 1.04i)4-s + (−0.275 + 0.623i)5-s + (0.337 − 0.576i)6-s + (−1.52 − 0.855i)7-s + (−0.170 − 0.0427i)8-s + (−0.780 + 0.119i)9-s + (0.517 + 0.846i)10-s + (0.510 − 0.191i)11-s + (−0.216 − 0.466i)12-s + (−0.628 − 1.12i)13-s + (−2.28 + 1.11i)14-s + (−0.104 + 0.294i)15-s + (0.650 − 0.569i)16-s + (−0.110 − 0.206i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 997 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.166397 + 1.44838i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.166397 + 1.44838i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 997 | \( 1 + (-31.4 + 2.82i)T \) |
| good | 2 | \( 1 + (-1.17 + 1.69i)T + (-0.703 - 1.87i)T^{2} \) |
| 3 | \( 1 + (-0.792 + 0.0601i)T + (2.96 - 0.452i)T^{2} \) |
| 5 | \( 1 + (0.616 - 1.39i)T + (-3.36 - 3.69i)T^{2} \) |
| 7 | \( 1 + (4.03 + 2.26i)T + (3.65 + 5.97i)T^{2} \) |
| 11 | \( 1 + (-1.69 + 0.635i)T + (8.27 - 7.24i)T^{2} \) |
| 13 | \( 1 + (2.26 + 4.04i)T + (-6.78 + 11.0i)T^{2} \) |
| 17 | \( 1 + (0.456 + 0.852i)T + (-9.41 + 14.1i)T^{2} \) |
| 19 | \( 1 + (4.10 + 3.59i)T + (2.50 + 18.8i)T^{2} \) |
| 23 | \( 1 + (3.55 + 6.33i)T + (-11.9 + 19.6i)T^{2} \) |
| 29 | \( 1 + (4.65 - 2.38i)T + (16.9 - 23.5i)T^{2} \) |
| 31 | \( 1 + (-8.27 - 4.84i)T + (15.1 + 27.0i)T^{2} \) |
| 37 | \( 1 + (-6.49 + 5.90i)T + (3.49 - 36.8i)T^{2} \) |
| 41 | \( 1 + (1.20 + 1.48i)T + (-8.47 + 40.1i)T^{2} \) |
| 43 | \( 1 + (6.47 - 6.85i)T + (-2.44 - 42.9i)T^{2} \) |
| 47 | \( 1 + (1.82 + 1.21i)T + (18.1 + 43.3i)T^{2} \) |
| 53 | \( 1 + (-6.98 - 2.18i)T + (43.5 + 30.1i)T^{2} \) |
| 59 | \( 1 + (1.80 + 2.94i)T + (-26.8 + 52.5i)T^{2} \) |
| 61 | \( 1 + (-4.85 - 10.9i)T + (-41.0 + 45.1i)T^{2} \) |
| 67 | \( 1 + (1.24 + 13.0i)T + (-65.8 + 12.6i)T^{2} \) |
| 71 | \( 1 + (-2.19 - 0.871i)T + (51.6 + 48.7i)T^{2} \) |
| 73 | \( 1 + (1.46 + 5.03i)T + (-61.5 + 39.2i)T^{2} \) |
| 79 | \( 1 + (-6.84 + 0.780i)T + (76.9 - 17.7i)T^{2} \) |
| 83 | \( 1 + (1.09 - 2.35i)T + (-53.4 - 63.4i)T^{2} \) |
| 89 | \( 1 + (-8.16 + 7.71i)T + (5.05 - 88.8i)T^{2} \) |
| 97 | \( 1 + (-1.19 + 3.59i)T + (-77.6 - 58.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.919689925595629077469954434859, −8.930912421910346375655816461203, −7.82601838358476383022963603876, −6.88208023959024190463091066028, −6.06732521585020718234163626021, −4.75462496851199559392207917987, −3.74564392446866926233289210344, −3.04914906222586354378479229360, −2.52512442299585646654216156139, −0.44648026451099292427546784013,
2.25273064416455753115021043038, 3.60547885729168047466971767993, 4.27508261437046456696079470618, 5.45956874967424385099758296353, 6.24017171705074956800721975005, 6.68223524138501030566015598234, 7.924792595103436744454048044701, 8.558584041270427161614125949331, 9.422551244075665187882055706349, 9.985088784754314232414366540514
