| L(s) = 1 | + (−0.752 + 2.24i)2-s + (−1.39 − 0.782i)3-s + (−2.89 − 2.18i)4-s + (0.987 − 1.48i)5-s + (2.80 − 2.54i)6-s + (−0.299 + 0.661i)7-s + (3.17 − 2.17i)8-s + (−0.233 − 0.384i)9-s + (2.60 + 3.34i)10-s + (−4.39 − 0.281i)11-s + (2.32 + 5.30i)12-s + (−4.39 − 5.07i)13-s + (−1.26 − 1.17i)14-s + (−2.54 + 1.30i)15-s + (0.516 + 1.80i)16-s + (1.19 − 0.526i)17-s + ⋯ |
| L(s) = 1 | + (−0.532 + 1.59i)2-s + (−0.803 − 0.451i)3-s + (−1.44 − 1.09i)4-s + (0.441 − 0.666i)5-s + (1.14 − 1.03i)6-s + (−0.113 + 0.250i)7-s + (1.12 − 0.770i)8-s + (−0.0779 − 0.128i)9-s + (0.824 + 1.05i)10-s + (−1.32 − 0.0849i)11-s + (0.669 + 1.53i)12-s + (−1.21 − 1.40i)13-s + (−0.337 − 0.313i)14-s + (−0.655 + 0.336i)15-s + (0.129 + 0.452i)16-s + (0.289 − 0.127i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.123690 + 0.408033i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.123690 + 0.408033i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 + (-11.1 - 29.3i)T \) |
| good | 2 | \( 1 + (0.752 - 2.24i)T + (-1.59 - 1.20i)T^{2} \) |
| 3 | \( 1 + (1.39 + 0.782i)T + (1.56 + 2.56i)T^{2} \) |
| 5 | \( 1 + (-0.987 + 1.48i)T + (-1.94 - 4.60i)T^{2} \) |
| 7 | \( 1 + (0.299 - 0.661i)T + (-4.61 - 5.26i)T^{2} \) |
| 11 | \( 1 + (4.39 + 0.281i)T + (10.9 + 1.40i)T^{2} \) |
| 13 | \( 1 + (4.39 + 5.07i)T + (-1.86 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.19 + 0.526i)T + (11.4 - 12.5i)T^{2} \) |
| 19 | \( 1 + (-3.98 - 3.04i)T + (4.98 + 18.3i)T^{2} \) |
| 23 | \( 1 + (-0.588 - 5.55i)T + (-22.4 + 4.82i)T^{2} \) |
| 29 | \( 1 + (3.23 - 4.68i)T + (-10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (-1.82 - 2.71i)T + (-11.7 + 28.7i)T^{2} \) |
| 37 | \( 1 + (-4.26 + 1.05i)T + (32.7 - 17.2i)T^{2} \) |
| 41 | \( 1 + (-3.39 - 0.305i)T + (40.3 + 7.30i)T^{2} \) |
| 43 | \( 1 + (-4.06 - 5.57i)T + (-13.1 + 40.9i)T^{2} \) |
| 47 | \( 1 + (3.46 + 11.5i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (4.46 - 4.09i)T + (4.57 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-11.8 - 0.909i)T + (58.3 + 9.02i)T^{2} \) |
| 61 | \( 1 + (6.12 + 5.19i)T + (9.90 + 60.1i)T^{2} \) |
| 67 | \( 1 + (4.34 - 3.73i)T + (10.0 - 66.2i)T^{2} \) |
| 71 | \( 1 + (2.00 - 1.64i)T + (14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (-12.1 - 8.66i)T + (23.6 + 69.0i)T^{2} \) |
| 79 | \( 1 + (-6.03 + 0.894i)T + (75.6 - 22.9i)T^{2} \) |
| 83 | \( 1 + (11.8 - 3.66i)T + (68.4 - 46.9i)T^{2} \) |
| 89 | \( 1 + (6.34 + 4.17i)T + (35.1 + 81.7i)T^{2} \) |
| 97 | \( 1 + (-1.33 + 0.0682i)T + (96.4 - 9.91i)T^{2} \) |
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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
−9.891845486480257646132667634130, −9.453477519251578254594659535112, −8.386885209024725737686091862679, −7.61301486466755554366726904139, −7.16141655890572705672433541719, −5.85940176131081722446142455877, −5.40472689026540835410923766148, −5.13599429970804824523483302117, −2.99195662265801772145187504655, −0.958327708589661091133730045537,
0.33024558072273003186873631090, 2.30005089705482215258174679795, 2.70972697805760686435939733683, 4.25813889850551646918643620769, 4.93435035903355794815314408452, 6.11746866249583915351340529904, 7.24255486949287945195327667304, 8.212586210324755193826904681698, 9.526343346248190634690063721851, 9.801021198824226919767394857803
