| L(s) = 1 | + (−0.696 + 2.07i)2-s + (−2.48 − 1.39i)3-s + (−2.24 − 1.69i)4-s + (1.99 − 3.00i)5-s + (4.62 − 4.18i)6-s + (1.53 − 3.39i)7-s + (1.46 − 1.00i)8-s + (2.65 + 4.35i)9-s + (4.85 + 6.23i)10-s + (−1.68 − 0.108i)11-s + (3.20 + 7.31i)12-s + (3.96 + 4.58i)13-s + (5.98 + 5.55i)14-s + (−9.12 + 4.67i)15-s + (−0.475 − 1.66i)16-s + (2.61 − 1.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.492 + 1.47i)2-s + (−1.43 − 0.804i)3-s + (−1.12 − 0.845i)4-s + (0.890 − 1.34i)5-s + (1.88 − 1.70i)6-s + (0.580 − 1.28i)7-s + (0.516 − 0.354i)8-s + (0.883 + 1.45i)9-s + (1.53 + 1.97i)10-s + (−0.509 − 0.0326i)11-s + (0.925 + 2.11i)12-s + (1.10 + 1.27i)13-s + (1.59 + 1.48i)14-s + (−2.35 + 1.20i)15-s + (−0.118 − 0.416i)16-s + (0.635 − 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.890087 - 0.217203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.890087 - 0.217203i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 + (14.2 + 27.9i)T \) |
| good | 2 | \( 1 + (0.696 - 2.07i)T + (-1.59 - 1.20i)T^{2} \) |
| 3 | \( 1 + (2.48 + 1.39i)T + (1.56 + 2.56i)T^{2} \) |
| 5 | \( 1 + (-1.99 + 3.00i)T + (-1.94 - 4.60i)T^{2} \) |
| 7 | \( 1 + (-1.53 + 3.39i)T + (-4.61 - 5.26i)T^{2} \) |
| 11 | \( 1 + (1.68 + 0.108i)T + (10.9 + 1.40i)T^{2} \) |
| 13 | \( 1 + (-3.96 - 4.58i)T + (-1.86 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.61 + 1.15i)T + (11.4 - 12.5i)T^{2} \) |
| 19 | \( 1 + (-5.19 - 3.97i)T + (4.98 + 18.3i)T^{2} \) |
| 23 | \( 1 + (0.368 + 3.47i)T + (-22.4 + 4.82i)T^{2} \) |
| 29 | \( 1 + (-4.78 + 6.92i)T + (-10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (-2.90 - 4.32i)T + (-11.7 + 28.7i)T^{2} \) |
| 37 | \( 1 + (0.942 - 0.233i)T + (32.7 - 17.2i)T^{2} \) |
| 41 | \( 1 + (-11.9 - 1.07i)T + (40.3 + 7.30i)T^{2} \) |
| 43 | \( 1 + (1.14 + 1.56i)T + (-13.1 + 40.9i)T^{2} \) |
| 47 | \( 1 + (2.01 + 6.71i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (1.57 - 1.44i)T + (4.57 - 52.8i)T^{2} \) |
| 59 | \( 1 + (10.1 + 0.779i)T + (58.3 + 9.02i)T^{2} \) |
| 61 | \( 1 + (7.22 + 6.13i)T + (9.90 + 60.1i)T^{2} \) |
| 67 | \( 1 + (5.31 - 4.56i)T + (10.0 - 66.2i)T^{2} \) |
| 71 | \( 1 + (2.38 - 1.94i)T + (14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (5.98 + 4.27i)T + (23.6 + 69.0i)T^{2} \) |
| 79 | \( 1 + (12.2 - 1.81i)T + (75.6 - 22.9i)T^{2} \) |
| 83 | \( 1 + (-14.9 + 4.62i)T + (68.4 - 46.9i)T^{2} \) |
| 89 | \( 1 + (-6.07 - 4.00i)T + (35.1 + 81.7i)T^{2} \) |
| 97 | \( 1 + (-2.88 + 0.147i)T + (96.4 - 9.91i)T^{2} \) |
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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
−9.816462868654180816179807135782, −8.846394006174590361797415763911, −7.940576727927905592167749367519, −7.39671446536165636622275353176, −6.34262327102421230224828657194, −5.96676183946685109314142867434, −5.05495325534019799913567385544, −4.46870948583208288156714388757, −1.45277805741795893376188960802, −0.77171253188310244219765354878,
1.23335262352974474444894731054, 2.72363126107349603710927069936, 3.27848740054657351992118038624, 4.86717053279865038853739344730, 5.82468174034721594912394101846, 6.09408401034532664835882161639, 7.68167561918477324231606084943, 8.964676887616485610906954596963, 9.679889840057814034007119958798, 10.39223565486503273410285512658
