| L(s) = 1 | + (0.571 − 1.65i)2-s + (0.458 + 0.888i)3-s + (−0.830 − 0.652i)4-s + (3.21 − 0.306i)5-s + (1.73 − 0.248i)6-s + (0.908 − 2.48i)7-s + (1.38 − 0.892i)8-s + (−0.580 + 0.814i)9-s + (1.32 − 5.48i)10-s + (−5.29 + 1.83i)11-s + (0.199 − 1.03i)12-s + (1.40 + 4.78i)13-s + (−3.58 − 2.92i)14-s + (1.74 + 2.71i)15-s + (−1.17 − 4.85i)16-s + (2.93 + 1.17i)17-s + ⋯ |
| L(s) = 1 | + (0.404 − 1.16i)2-s + (0.264 + 0.513i)3-s + (−0.415 − 0.326i)4-s + (1.43 − 0.137i)5-s + (0.706 − 0.101i)6-s + (0.343 − 0.939i)7-s + (0.490 − 0.315i)8-s + (−0.193 + 0.271i)9-s + (0.420 − 1.73i)10-s + (−1.59 + 0.552i)11-s + (0.0576 − 0.299i)12-s + (0.389 + 1.32i)13-s + (−0.958 − 0.780i)14-s + (0.450 + 0.700i)15-s + (−0.294 − 1.21i)16-s + (0.712 + 0.285i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.02362 - 1.34632i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02362 - 1.34632i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.458 - 0.888i)T \) |
| 7 | \( 1 + (-0.908 + 2.48i)T \) |
| 23 | \( 1 + (4.49 + 1.67i)T \) |
| good | 2 | \( 1 + (-0.571 + 1.65i)T + (-1.57 - 1.23i)T^{2} \) |
| 5 | \( 1 + (-3.21 + 0.306i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (5.29 - 1.83i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (-1.40 - 4.78i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-2.93 - 1.17i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (1.34 - 0.538i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (0.946 + 6.58i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (4.62 - 0.220i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-4.74 - 3.37i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (3.80 + 1.73i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (3.23 - 5.03i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-8.59 + 4.96i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.30 - 2.41i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (6.20 + 1.50i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (2.11 + 1.09i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (-0.499 - 2.59i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (-11.0 - 12.6i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (5.15 - 6.54i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (5.87 - 6.15i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (5.96 + 13.0i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (0.220 - 4.63i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (1.13 - 2.49i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.65786043182094723738734361162, −10.04618889217135823892667048035, −9.670611618498676402043713557749, −8.254103463857602393931658857051, −7.19127183618570970004630941709, −5.83286597618189141364299153115, −4.73117937288253036782614987796, −3.93631802931318669702612148416, −2.48895891294454001852308818618, −1.70229841829249141825983215187,
1.92107070123480622780786101264, 2.97849479914848194033379685419, 5.29742722914544117169884997090, 5.54974010685779979905658483545, 6.22737602919636450845151202486, 7.55775545930090772921612581603, 8.126678206859653528828759884566, 9.073394005008424049216954313683, 10.30708812449084537287602274656, 10.92435655592617471748357015534
