| L(s) = 1 | + (−0.555 + 1.65i)2-s + (0.352 + 0.198i)3-s + (−0.848 − 0.640i)4-s + (−0.636 + 0.960i)5-s + (−0.525 + 0.475i)6-s + (−1.20 + 2.66i)7-s + (−1.35 + 0.927i)8-s + (−1.47 − 2.42i)9-s + (−1.24 − 1.59i)10-s + (2.05 + 0.131i)11-s + (−0.172 − 0.394i)12-s + (−4.41 − 5.10i)13-s + (−3.74 − 3.48i)14-s + (−0.415 + 0.212i)15-s + (−1.37 − 4.80i)16-s + (3.75 − 1.65i)17-s + ⋯ |
| L(s) = 1 | + (−0.392 + 1.17i)2-s + (0.203 + 0.114i)3-s + (−0.424 − 0.320i)4-s + (−0.284 + 0.429i)5-s + (−0.214 + 0.194i)6-s + (−0.456 + 1.00i)7-s + (−0.478 + 0.328i)8-s + (−0.491 − 0.807i)9-s + (−0.392 − 0.502i)10-s + (0.618 + 0.0396i)11-s + (−0.0498 − 0.113i)12-s + (−1.22 − 1.41i)13-s + (−1.00 − 0.930i)14-s + (−0.107 + 0.0549i)15-s + (−0.343 − 1.20i)16-s + (0.911 − 0.402i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.120460 - 0.0633121i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.120460 - 0.0633121i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 + (-0.507 + 31.3i)T \) |
| good | 2 | \( 1 + (0.555 - 1.65i)T + (-1.59 - 1.20i)T^{2} \) |
| 3 | \( 1 + (-0.352 - 0.198i)T + (1.56 + 2.56i)T^{2} \) |
| 5 | \( 1 + (0.636 - 0.960i)T + (-1.94 - 4.60i)T^{2} \) |
| 7 | \( 1 + (1.20 - 2.66i)T + (-4.61 - 5.26i)T^{2} \) |
| 11 | \( 1 + (-2.05 - 0.131i)T + (10.9 + 1.40i)T^{2} \) |
| 13 | \( 1 + (4.41 + 5.10i)T + (-1.86 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.75 + 1.65i)T + (11.4 - 12.5i)T^{2} \) |
| 19 | \( 1 + (5.73 + 4.38i)T + (4.98 + 18.3i)T^{2} \) |
| 23 | \( 1 + (0.323 + 3.05i)T + (-22.4 + 4.82i)T^{2} \) |
| 29 | \( 1 + (3.34 - 4.84i)T + (-10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (-2.65 - 3.94i)T + (-11.7 + 28.7i)T^{2} \) |
| 37 | \( 1 + (6.49 - 1.61i)T + (32.7 - 17.2i)T^{2} \) |
| 41 | \( 1 + (1.78 + 0.160i)T + (40.3 + 7.30i)T^{2} \) |
| 43 | \( 1 + (-5.46 - 7.49i)T + (-13.1 + 40.9i)T^{2} \) |
| 47 | \( 1 + (3.10 + 10.3i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-3.35 + 3.07i)T + (4.57 - 52.8i)T^{2} \) |
| 59 | \( 1 + (14.4 + 1.11i)T + (58.3 + 9.02i)T^{2} \) |
| 61 | \( 1 + (-4.78 - 4.05i)T + (9.90 + 60.1i)T^{2} \) |
| 67 | \( 1 + (-8.82 + 7.59i)T + (10.0 - 66.2i)T^{2} \) |
| 71 | \( 1 + (1.37 - 1.12i)T + (14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (7.43 + 5.31i)T + (23.6 + 69.0i)T^{2} \) |
| 79 | \( 1 + (9.65 - 1.43i)T + (75.6 - 22.9i)T^{2} \) |
| 83 | \( 1 + (-4.71 + 1.46i)T + (68.4 - 46.9i)T^{2} \) |
| 89 | \( 1 + (14.8 + 9.80i)T + (35.1 + 81.7i)T^{2} \) |
| 97 | \( 1 + (8.30 - 0.425i)T + (96.4 - 9.91i)T^{2} \) |
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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
−9.487770732904743744239250142760, −8.875617971117456991793279125985, −8.248442303895435695122519687650, −7.20912827049244701344740602169, −6.64340433924545365636424969134, −5.76330587048386537809319022232, −4.99854321820325643743960689806, −3.24335346107419652158616959599, −2.71688807490381816183240360002, −0.06646936347609322239551910757,
1.52684259469042637291850532146, 2.45722293935804566472723102935, 3.82098852031421964808364296223, 4.33707226714673872296886241509, 5.87013791905880907393978401412, 6.87382852660277750102742437496, 7.78917988111521356008574685706, 8.676641788721114301369081416867, 9.555744532721729009734128211683, 10.13598000548943634286957777319
