L(s) = 1 | + 1.27i·2-s + i·3-s + 0.384·4-s + (1.01 + 1.99i)5-s − 1.27·6-s + 0.483i·7-s + 3.03i·8-s − 9-s + (−2.53 + 1.28i)10-s + 0.384i·12-s − 3.14i·13-s − 0.614·14-s + (−1.99 + 1.01i)15-s − 3.08·16-s − 2.51i·17-s − 1.27i·18-s + ⋯ |
L(s) = 1 | + 0.898i·2-s + 0.577i·3-s + 0.192·4-s + (0.453 + 0.891i)5-s − 0.518·6-s + 0.182i·7-s + 1.07i·8-s − 0.333·9-s + (−0.801 + 0.407i)10-s + 0.111i·12-s − 0.872i·13-s − 0.164·14-s + (−0.514 + 0.261i)15-s − 0.770·16-s − 0.608i·17-s − 0.299i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.632718751\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.632718751\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1.01 - 1.99i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.27iT - 2T^{2} \) |
| 7 | \( 1 - 0.483iT - 7T^{2} \) |
| 13 | \( 1 + 3.14iT - 13T^{2} \) |
| 17 | \( 1 + 2.51iT - 17T^{2} \) |
| 19 | \( 1 + 3.72T + 19T^{2} \) |
| 23 | \( 1 - 8.53iT - 23T^{2} \) |
| 29 | \( 1 + 9.98T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 + 3.20iT - 37T^{2} \) |
| 41 | \( 1 - 7.86T + 41T^{2} \) |
| 43 | \( 1 - 5.97iT - 43T^{2} \) |
| 47 | \( 1 - 6.13iT - 47T^{2} \) |
| 53 | \( 1 - 1.58iT - 53T^{2} \) |
| 59 | \( 1 - 0.377T + 59T^{2} \) |
| 61 | \( 1 + 3.00T + 61T^{2} \) |
| 67 | \( 1 - 5.98iT - 67T^{2} \) |
| 71 | \( 1 - 8.34T + 71T^{2} \) |
| 73 | \( 1 - 0.151iT - 73T^{2} \) |
| 79 | \( 1 - 7.48T + 79T^{2} \) |
| 83 | \( 1 + 4.05iT - 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 3.60iT - 97T^{2} \) |
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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
−9.490327921272002464900656103164, −9.070104491481453437883796187458, −7.64504858936953761856101783863, −7.58451401368634449922187797820, −6.45598179963736064754717636370, −5.67821451100175414250341985790, −5.32616921178396085297730606877, −3.88508803940962631563180112478, −2.93089938672629379144740292191, −1.99421733611488499017889328086,
0.54384407581925886299424281002, 1.84663638806089236219387045953, 2.23011269518760088575006300330, 3.72474834572379537010009976787, 4.43355326740789072569710113500, 5.65780543293106493020973468229, 6.44814419267189688939463917323, 7.13798078099422930065217339064, 8.174954481509282601218897069874, 8.982940567398907871689888943313