L(s) = 1 | + (−1.67 + 0.965i)5-s + (0.633 + 0.366i)7-s + (−2.63 + 4.57i)11-s + (−2.23 − 3.86i)13-s + 0.896i·17-s − 1.26i·19-s + (−0.707 − 1.22i)23-s + (−0.633 + 1.09i)25-s + (4.69 + 2.70i)29-s + (6.46 − 3.73i)31-s − 1.41·35-s − 7.73·37-s + (0.328 − 0.189i)41-s + (−7.56 − 4.36i)43-s + (2.31 − 4.00i)47-s + ⋯ |
L(s) = 1 | + (−0.748 + 0.431i)5-s + (0.239 + 0.138i)7-s + (−0.795 + 1.37i)11-s + (−0.619 − 1.07i)13-s + 0.217i·17-s − 0.290i·19-s + (−0.147 − 0.255i)23-s + (−0.126 + 0.219i)25-s + (0.871 + 0.502i)29-s + (1.16 − 0.670i)31-s − 0.239·35-s − 1.27·37-s + (0.0512 − 0.0295i)41-s + (−1.15 − 0.665i)43-s + (0.337 − 0.583i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5710973038\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5710973038\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.67 - 0.965i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.633 - 0.366i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.63 - 4.57i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.23 + 3.86i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 0.896iT - 17T^{2} \) |
| 19 | \( 1 + 1.26iT - 19T^{2} \) |
| 23 | \( 1 + (0.707 + 1.22i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.69 - 2.70i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.46 + 3.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.73T + 37T^{2} \) |
| 41 | \( 1 + (-0.328 + 0.189i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (7.56 + 4.36i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.31 + 4.00i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 2.44iT - 53T^{2} \) |
| 59 | \( 1 + (5.41 + 9.38i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.598 + 1.03i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.3 + 6.56i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + (-14.3 - 8.29i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.27 - 9.14i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 14.9iT - 89T^{2} \) |
| 97 | \( 1 + (-3.19 + 5.53i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.333245663978546893400849479167, −7.996525864017959625148599985489, −7.16776570632018377592878405845, −6.61830083814435569467819712063, −5.25211918144022756321367448718, −4.89635723904019646757603379660, −3.79168974826386941449670383214, −2.87124038128430749271498230477, −1.97095696990355508798869287563, −0.20775149752520766892665231550,
1.09417470922281486472975426559, 2.51739347473293342513889045653, 3.45329595377975553460451565254, 4.43433159552770523141132522689, 5.01308338169156326486436157064, 6.03148353562391242418365052433, 6.82022833182690401021821010621, 7.80016912950278258995805875751, 8.256127829664593976770241900170, 8.899067411538674329274895700806