L(s) = 1 | − 0.708·2-s + (−0.837 − 1.45i)3-s − 1.49·4-s + (1.10 − 1.90i)5-s + (0.593 + 1.02i)6-s + (1.50 + 2.59i)7-s + 2.47·8-s + (0.0955 − 0.165i)9-s + (−0.779 + 1.35i)10-s + (2.27 − 3.94i)11-s + (1.25 + 2.17i)12-s + (−0.5 + 0.866i)13-s + (−1.06 − 1.84i)14-s − 3.68·15-s + 1.23·16-s + (−3.32 − 5.76i)17-s + ⋯ |
L(s) = 1 | − 0.501·2-s + (−0.483 − 0.837i)3-s − 0.748·4-s + (0.492 − 0.852i)5-s + (0.242 + 0.419i)6-s + (0.567 + 0.982i)7-s + 0.876·8-s + (0.0318 − 0.0551i)9-s + (−0.246 + 0.427i)10-s + (0.686 − 1.18i)11-s + (0.362 + 0.627i)12-s + (−0.138 + 0.240i)13-s + (−0.284 − 0.492i)14-s − 0.952·15-s + 0.309·16-s + (−0.806 − 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.341347 - 0.679419i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.341347 - 0.679419i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.0357 + 5.56i)T \) |
good | 2 | \( 1 + 0.708T + 2T^{2} \) |
| 3 | \( 1 + (0.837 + 1.45i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.10 + 1.90i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.50 - 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.27 + 3.94i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.32 + 5.76i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.733 - 1.26i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 6.58T + 23T^{2} \) |
| 29 | \( 1 + 5.25T + 29T^{2} \) |
| 37 | \( 1 + (-1.42 - 2.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.27 + 5.66i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.10 - 1.90i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + (-5.14 + 8.91i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.65 - 4.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + (0.702 - 1.21i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.94 + 8.56i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.60 + 2.77i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.38 - 2.39i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.35 - 12.7i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 8.89T + 89T^{2} \) |
| 97 | \( 1 + 0.237T + 97T^{2} \) |
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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
−11.18322966975924346392548274246, −9.595309483754429846312628734496, −9.116108850619539035508304864923, −8.391790463454130156570814065960, −7.34692242051489354518610849461, −5.97340737602969305962999889278, −5.38158413743439815088905261635, −4.11866004065963831669485722660, −1.94973495824527024682038609963, −0.67114498117019022897724990599,
1.79617243844052592378229575332, 4.09125545732207831109072236749, 4.41320068163002811989756056711, 5.76876663546926684216577026139, 7.05345956898963220463423099916, 7.87202437650582961788493274621, 9.135030659940491978303624819923, 10.00488707425165046572418044415, 10.50088031773377643560526893914, 11.00540752821560737056255093169