L(s) = 1 | + (0.800 + 1.16i)2-s + i·3-s + (−0.719 + 1.86i)4-s + i·5-s + (−1.16 + 0.800i)6-s − 1.17·7-s + (−2.75 + 0.653i)8-s − 9-s + (−1.16 + 0.800i)10-s + 4.08·11-s + (−1.86 − 0.719i)12-s − 5.99·13-s + (−0.942 − 1.37i)14-s − 15-s + (−2.96 − 2.68i)16-s + 0.761i·17-s + ⋯ |
L(s) = 1 | + (0.565 + 0.824i)2-s + 0.577i·3-s + (−0.359 + 0.932i)4-s + 0.447i·5-s + (−0.476 + 0.326i)6-s − 0.445·7-s + (−0.972 + 0.231i)8-s − 0.333·9-s + (−0.368 + 0.253i)10-s + 1.23·11-s + (−0.538 − 0.207i)12-s − 1.66·13-s + (−0.252 − 0.367i)14-s − 0.258·15-s + (−0.740 − 0.671i)16-s + 0.184i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8837494593\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8837494593\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.800 - 1.16i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-4.78 - 0.308i)T \) |
good | 7 | \( 1 + 1.17T + 7T^{2} \) |
| 11 | \( 1 - 4.08T + 11T^{2} \) |
| 13 | \( 1 + 5.99T + 13T^{2} \) |
| 17 | \( 1 - 0.761iT - 17T^{2} \) |
| 19 | \( 1 + 6.34T + 19T^{2} \) |
| 29 | \( 1 + 6.79T + 29T^{2} \) |
| 31 | \( 1 - 1.55iT - 31T^{2} \) |
| 37 | \( 1 - 1.24iT - 37T^{2} \) |
| 41 | \( 1 + 1.89T + 41T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 - 2.61iT - 47T^{2} \) |
| 53 | \( 1 - 2.04iT - 53T^{2} \) |
| 59 | \( 1 - 1.92iT - 59T^{2} \) |
| 61 | \( 1 + 4.50iT - 61T^{2} \) |
| 67 | \( 1 - 5.89T + 67T^{2} \) |
| 71 | \( 1 - 9.16iT - 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 7.39T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 12.5iT - 89T^{2} \) |
| 97 | \( 1 - 5.05iT - 97T^{2} \) |
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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
−9.861860862631045215279797829399, −9.293085886013460315991058289777, −8.537492447296355735921857702698, −7.43808244032963556345670370989, −6.79666492234354450741154233294, −6.09964514640389794233148223910, −5.07297647378340949543576428530, −4.26374014159712017311464828589, −3.46577754404134492689909472224, −2.40777378544637255926669077096,
0.27978132219006893115516301933, 1.70205062349675021501567609405, 2.62456860093510037608429855090, 3.80636075856628892907920836806, 4.66081045885306500498339236629, 5.56259730067602903052690236699, 6.53222807856333344849890958907, 7.15973949831015978399681460830, 8.428639149493183496901552562815, 9.327277296416951527891739796109