L(s) = 1 | + (−0.5 + 0.133i)3-s + (−2 − i)5-s + (−2.13 + 1.23i)7-s + (−2.36 + 1.36i)9-s + (−1.13 − 4.23i)11-s + (−2 + 3i)13-s + (1.13 + 0.232i)15-s + (−0.232 + 0.866i)17-s + (−2.86 − 0.767i)19-s + (0.901 − 0.901i)21-s + (−0.0358 − 0.133i)23-s + (3 + 4i)25-s + (2.09 − 2.09i)27-s + (−1.5 − 0.866i)29-s + (−5.19 − 5.19i)31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.0773i)3-s + (−0.894 − 0.447i)5-s + (−0.806 + 0.465i)7-s + (−0.788 + 0.455i)9-s + (−0.341 − 1.27i)11-s + (−0.554 + 0.832i)13-s + (0.292 + 0.0599i)15-s + (−0.0562 + 0.210i)17-s + (−0.657 − 0.176i)19-s + (0.196 − 0.196i)21-s + (−0.00748 − 0.0279i)23-s + (0.600 + 0.800i)25-s + (0.403 − 0.403i)27-s + (−0.278 − 0.160i)29-s + (−0.933 − 0.933i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 13 | \( 1 + (2 - 3i)T \) |
good | 3 | \( 1 + (0.5 - 0.133i)T + (2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (2.13 - 1.23i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.13 + 4.23i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (0.232 - 0.866i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (2.86 + 0.767i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (0.0358 + 0.133i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (1.5 + 0.866i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.19 + 5.19i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.33 - 0.767i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.33 + 2.5i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (5.96 + 1.59i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 - 10.9iT - 47T^{2} \) |
| 53 | \( 1 + (-2.46 - 2.46i)T + 53iT^{2} \) |
| 59 | \( 1 + (-2.33 + 8.69i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.13 + 10.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.598 + 2.23i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 0.535iT - 79T^{2} \) |
| 83 | \( 1 - 2.92iT - 83T^{2} \) |
| 89 | \( 1 + (14.7 - 3.96i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-3.86 - 6.69i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.40930939569602749164264107440, −10.91016409936881307445572337498, −9.413793070350809790817696357432, −8.631448819359183117117685580304, −7.71520440852332289957240188249, −6.33284359158995376358876461607, −5.40279072283639638831662990651, −4.08168167632930616719058901640, −2.73706155896064975018308445461, 0,
2.80424179502574274612603317615, 3.97357922874094054930924309737, 5.34669825879048763921914900698, 6.72843737924971687083652450378, 7.36505334215355171349089517679, 8.515298786171705170278162602288, 9.814470120017475705842325128333, 10.55781865812136319524366306394, 11.56704897213737104441459579286