L(s) = 1 | + (−0.564 + 1.73i)2-s + (0.978 + 0.207i)3-s + (−1.89 − 1.37i)4-s + (0.5 + 0.866i)5-s + (−0.913 + 1.58i)6-s + (1.97 − 1.43i)8-s + (0.913 + 0.406i)9-s + (−1.78 + 0.379i)10-s + (−1.56 − 1.73i)12-s + (0.309 + 0.951i)15-s + (0.657 + 2.02i)16-s + (−0.104 − 0.994i)17-s + (−1.22 + 1.35i)18-s + (−1.30 − 1.45i)19-s + (0.244 − 2.32i)20-s + ⋯ |
L(s) = 1 | + (−0.564 + 1.73i)2-s + (0.978 + 0.207i)3-s + (−1.89 − 1.37i)4-s + (0.5 + 0.866i)5-s + (−0.913 + 1.58i)6-s + (1.97 − 1.43i)8-s + (0.913 + 0.406i)9-s + (−1.78 + 0.379i)10-s + (−1.56 − 1.73i)12-s + (0.309 + 0.951i)15-s + (0.657 + 2.02i)16-s + (−0.104 − 0.994i)17-s + (−1.22 + 1.35i)18-s + (−1.30 − 1.45i)19-s + (0.244 − 2.32i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8512452116\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8512452116\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.978 - 0.207i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
good | 2 | \( 1 + (0.564 - 1.73i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 11 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 13 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 17 | \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
| 19 | \( 1 + (1.30 + 1.45i)T + (-0.104 + 0.994i)T^{2} \) |
| 23 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 43 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 47 | \( 1 + (-0.0646 - 0.198i)T + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.564 + 0.251i)T + (0.669 + 0.743i)T^{2} \) |
| 59 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 61 | \( 1 - 1.33T + T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 73 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 79 | \( 1 + (-0.204 - 1.94i)T + (-0.978 + 0.207i)T^{2} \) |
| 83 | \( 1 + (0.204 - 0.0434i)T + (0.913 - 0.406i)T^{2} \) |
| 89 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−11.28050085973178033868534486084, −10.11641276188211851364413516505, −9.561977614440646582839685244613, −8.789670798470460792323326417210, −7.87786352392975329356587695111, −7.10364997920070288779919919962, −6.41999012637080455826712328971, −5.25742250890246169110994947315, −4.07784348472268510782985739312, −2.40866207458106990006195841559,
1.54647328164570292722048344978, 2.31613845537286452056650186894, 3.75548932802332619620886991724, 4.44755270965318884367978486224, 6.20216752767361646757679751164, 8.059515437068500862489750220573, 8.443978150071307125080333628466, 9.188729417711295512716811164589, 10.21061161973832015572766633449, 10.45516902458461774172417097649