| 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
−9.985088784754314232414366540514, −9.422551244075665187882055706349, −8.558584041270427161614125949331, −7.924792595103436744454048044701, −6.68223524138501030566015598234, −6.24017171705074956800721975005, −5.45956874967424385099758296353, −4.27508261437046456696079470618, −3.60547885729168047466971767993, −2.25273064416455753115021043038,
0.44648026451099292427546784013, 2.52512442299585646654216156139, 3.04914906222586354378479229360, 3.74564392446866926233289210344, 4.75462496851199559392207917987, 6.06732521585020718234163626021, 6.88208023959024190463091066028, 7.82601838358476383022963603876, 8.930912421910346375655816461203, 9.919689925595629077469954434859
