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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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
−8.594211609367050483868319476040, −7.57428242640432274904725860634, −7.14551390844737301765096196207, −6.10551613172193044000590963780, −5.50075811956938708121346733064, −4.36553645727638714774948747390, −3.36842516248844923733316318269, −3.05483175674768847536828240366, −1.34608730887220555498278166134, 0,
1.61091724426819399199240911592, 2.59324060387441093548150311438, 3.99201350354711019370563099427, 4.36726687187711539005717844511, 5.21429232619716735215359709131, 6.40618469024043876695598735417, 7.07188082718504770366128073972, 7.62034503311180646598455225726, 8.645558478807293992993555488334