L(s) = 1 | + (0.449 − 1.34i)2-s + (−1.59 − 1.20i)4-s + i·5-s + (−1.40 + 2.24i)7-s + (−2.33 + 1.59i)8-s + (1.34 + 0.449i)10-s − 3.99i·11-s + 2.50i·13-s + (2.38 + 2.88i)14-s + (1.09 + 3.84i)16-s − 1.07i·17-s + 7.78·19-s + (1.20 − 1.59i)20-s + (−5.35 − 1.79i)22-s − 8.10i·23-s + ⋯ |
L(s) = 1 | + (0.317 − 0.948i)2-s + (−0.797 − 0.602i)4-s + 0.447i·5-s + (−0.529 + 0.848i)7-s + (−0.825 + 0.564i)8-s + (0.424 + 0.142i)10-s − 1.20i·11-s + 0.694i·13-s + (0.636 + 0.771i)14-s + (0.273 + 0.961i)16-s − 0.261i·17-s + 1.78·19-s + (0.269 − 0.356i)20-s + (−1.14 − 0.382i)22-s − 1.69i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.653715029\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.653715029\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.449 + 1.34i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (1.40 - 2.24i)T \) |
good | 11 | \( 1 + 3.99iT - 11T^{2} \) |
| 13 | \( 1 - 2.50iT - 13T^{2} \) |
| 17 | \( 1 + 1.07iT - 17T^{2} \) |
| 19 | \( 1 - 7.78T + 19T^{2} \) |
| 23 | \( 1 + 8.10iT - 23T^{2} \) |
| 29 | \( 1 - 5.99T + 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 - 3.05T + 37T^{2} \) |
| 41 | \( 1 - 9.32iT - 41T^{2} \) |
| 43 | \( 1 + 8.12iT - 43T^{2} \) |
| 47 | \( 1 - 9.01T + 47T^{2} \) |
| 53 | \( 1 + 2.86T + 53T^{2} \) |
| 59 | \( 1 - 9.68T + 59T^{2} \) |
| 61 | \( 1 + 3.21iT - 61T^{2} \) |
| 67 | \( 1 - 7.79iT - 67T^{2} \) |
| 71 | \( 1 - 5.84iT - 71T^{2} \) |
| 73 | \( 1 - 5.73iT - 73T^{2} \) |
| 79 | \( 1 - 2.81iT - 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 3.51iT - 89T^{2} \) |
| 97 | \( 1 + 12.0iT - 97T^{2} \) |
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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
−9.738391425160829025611051138441, −8.832782330055963184115767637483, −8.287157337933031358681691429920, −6.80786459264795170337682357131, −6.05357341395165311773769397834, −5.25960391909176343687400730436, −4.17588410044334244072246221407, −3.01478814852718280262367469800, −2.59464893076217140072063371514, −0.876863490092731376518864127700,
0.997558202580968321330242214000, 3.02790552339592033439826821286, 3.96344258946522134575071738803, 4.85525951490022348606682613160, 5.61654722109181005919582815098, 6.61176732182030530238004814744, 7.54269832747415232236254455697, 7.76245072602668884312973738774, 9.040127540484439732849043963286, 9.747821136415432022039355159683