L(s) = 1 | + (1.40 + 0.148i)2-s + (0.0994 + 0.182i)3-s + (1.95 + 0.418i)4-s + (0.0408 − 2.23i)5-s + (0.112 + 0.271i)6-s + (−2.09 − 2.80i)7-s + (2.68 + 0.880i)8-s + (1.59 − 2.48i)9-s + (0.390 − 3.13i)10-s + (−0.956 + 0.436i)11-s + (0.118 + 0.398i)12-s + (1.28 + 0.959i)13-s + (−2.53 − 4.25i)14-s + (0.411 − 0.214i)15-s + (3.64 + 1.63i)16-s + (−3.78 + 0.270i)17-s + ⋯ |
L(s) = 1 | + (0.994 + 0.105i)2-s + (0.0574 + 0.105i)3-s + (0.977 + 0.209i)4-s + (0.0182 − 0.999i)5-s + (0.0460 + 0.110i)6-s + (−0.793 − 1.06i)7-s + (0.950 + 0.311i)8-s + (0.532 − 0.829i)9-s + (0.123 − 0.992i)10-s + (−0.288 + 0.131i)11-s + (0.0341 + 0.114i)12-s + (0.355 + 0.266i)13-s + (−0.677 − 1.13i)14-s + (0.106 − 0.0555i)15-s + (0.912 + 0.409i)16-s + (−0.917 + 0.0656i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.37882 - 0.849731i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.37882 - 0.849731i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.148i)T \) |
| 5 | \( 1 + (-0.0408 + 2.23i)T \) |
| 23 | \( 1 + (-2.44 - 4.12i)T \) |
good | 3 | \( 1 + (-0.0994 - 0.182i)T + (-1.62 + 2.52i)T^{2} \) |
| 7 | \( 1 + (2.09 + 2.80i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (0.956 - 0.436i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-1.28 - 0.959i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (3.78 - 0.270i)T + (16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (-1.55 - 1.79i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-4.94 - 4.28i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.00306 - 0.0104i)T + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-2.06 - 9.49i)T + (-33.6 + 15.3i)T^{2} \) |
| 41 | \( 1 + (-3.82 + 2.45i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (2.54 + 4.66i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (6.89 - 6.89i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.984 - 1.31i)T + (-14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (0.811 + 5.64i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-0.228 + 0.0671i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (5.38 + 14.4i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (-0.804 - 0.367i)T + (46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-7.19 - 0.514i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (-1.38 - 9.66i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (0.867 + 3.98i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (1.49 - 5.10i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (14.8 + 3.23i)T + (88.2 + 40.2i)T^{2} \) |
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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
−11.10496518993858662853979867322, −10.10047261489940328164445049231, −9.295579004605908216120195777392, −8.065759661840338364469886859016, −6.97334812185332424722481782540, −6.32191607215406089956855463163, −5.00292408605892936714966445953, −4.14685045971370691173989553741, −3.29480228246526443955377402501, −1.31694749327550995679884726556,
2.33332918146672381061600814935, 2.93448199695928719922982625304, 4.33141420398397755980380586988, 5.53887681444443622316970580121, 6.41009520396653279310220852075, 7.13116975464577532434983266023, 8.260917695875488457717106811226, 9.615724791765294693516400456033, 10.55046746356373404618621867743, 11.17402433206456505820014040865