| L(s) = 1 | + (0.541 − 1.56i)2-s + (0.458 + 0.888i)3-s + (−0.578 − 0.455i)4-s + (−1.66 + 0.158i)5-s + (1.63 − 0.235i)6-s + (1.09 + 2.41i)7-s + (1.75 − 1.13i)8-s + (−0.580 + 0.814i)9-s + (−0.652 + 2.68i)10-s + (1.58 − 0.548i)11-s + (0.139 − 0.722i)12-s + (0.290 + 0.989i)13-s + (4.35 − 0.400i)14-s + (−0.903 − 1.40i)15-s + (−1.16 − 4.79i)16-s + (7.44 + 2.97i)17-s + ⋯ |
| L(s) = 1 | + (0.382 − 1.10i)2-s + (0.264 + 0.513i)3-s + (−0.289 − 0.227i)4-s + (−0.744 + 0.0710i)5-s + (0.668 − 0.0961i)6-s + (0.412 + 0.911i)7-s + (0.621 − 0.399i)8-s + (−0.193 + 0.271i)9-s + (−0.206 + 0.849i)10-s + (0.477 − 0.165i)11-s + (0.0402 − 0.208i)12-s + (0.0806 + 0.274i)13-s + (1.16 − 0.107i)14-s + (−0.233 − 0.363i)15-s + (−0.290 − 1.19i)16-s + (1.80 + 0.722i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.92603 - 0.397871i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.92603 - 0.397871i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.09 - 2.41i)T \) |
| 23 | \( 1 + (-4.47 - 1.71i)T \) |
| good | 2 | \( 1 + (-0.541 + 1.56i)T + (-1.57 - 1.23i)T^{2} \) |
| 5 | \( 1 + (1.66 - 0.158i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (-1.58 + 0.548i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (-0.290 - 0.989i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-7.44 - 2.97i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (1.17 - 0.471i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (1.03 + 7.22i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (0.732 - 0.0348i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-2.16 - 1.54i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (2.27 + 1.03i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (1.31 - 2.04i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (2.89 - 1.66i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.76 + 8.14i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-4.72 - 1.14i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (8.22 + 4.24i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (0.325 + 1.68i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (6.99 + 8.07i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (8.07 - 10.2i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (-2.74 + 2.88i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (1.39 + 3.05i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.110 + 2.32i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (-1.44 + 3.17i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.25075128090113723759997663413, −10.16546066761033619691742845960, −9.412805013129462509198329657098, −8.274310607334096953394827178448, −7.56589667044183082759037227530, −6.02055862169720304615317984662, −4.82218841096768997627737509715, −3.79961685519839654486524594360, −3.06976812972039692013046381642, −1.66575780465738771678054981283,
1.28489923825988038408601454450, 3.33402611173720852625491625304, 4.49541192468588879136254617289, 5.46501149068016815853845191180, 6.66452365382897997749386046301, 7.45945158734436805229754840590, 7.80897443280915117097824570287, 8.862277081576962475520764128421, 10.20856840814213072433182473818, 11.12946051829297906008216382099
