L(s) = 1 | + (2.05 + 2.05i)3-s + (−2.72 + 2.72i)5-s − i·7-s + 5.44i·9-s + (−0.919 + 0.919i)11-s + (1.12 + 1.12i)13-s − 11.2·15-s − 1.50·17-s + (−1.46 − 1.46i)19-s + (2.05 − 2.05i)21-s + 4.77i·23-s − 9.88i·25-s + (−5.02 + 5.02i)27-s + (−4.10 − 4.10i)29-s + 4.10·31-s + ⋯ |
L(s) = 1 | + (1.18 + 1.18i)3-s + (−1.21 + 1.21i)5-s − 0.377i·7-s + 1.81i·9-s + (−0.277 + 0.277i)11-s + (0.312 + 0.312i)13-s − 2.89·15-s − 0.365·17-s + (−0.335 − 0.335i)19-s + (0.448 − 0.448i)21-s + 0.994i·23-s − 1.97i·25-s + (−0.967 + 0.967i)27-s + (−0.761 − 0.761i)29-s + 0.738·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.206981 + 1.48646i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.206981 + 1.48646i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (-2.05 - 2.05i)T + 3iT^{2} \) |
| 5 | \( 1 + (2.72 - 2.72i)T - 5iT^{2} \) |
| 11 | \( 1 + (0.919 - 0.919i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.12 - 1.12i)T + 13iT^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 + (1.46 + 1.46i)T + 19iT^{2} \) |
| 23 | \( 1 - 4.77iT - 23T^{2} \) |
| 29 | \( 1 + (4.10 + 4.10i)T + 29iT^{2} \) |
| 31 | \( 1 - 4.10T + 31T^{2} \) |
| 37 | \( 1 + (-1.65 + 1.65i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.45iT - 41T^{2} \) |
| 43 | \( 1 + (5.68 - 5.68i)T - 43iT^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 + (-0.675 + 0.675i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.13 - 1.13i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3.21 - 3.21i)T + 61iT^{2} \) |
| 67 | \( 1 + (-1.52 - 1.52i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 + 14.4iT - 73T^{2} \) |
| 79 | \( 1 + 1.77T + 79T^{2} \) |
| 83 | \( 1 + (-7.16 - 7.16i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.45iT - 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−10.36821240822901363167791097493, −9.769241949687982084399046349688, −8.823948127342503241728254925758, −7.939617619684325204212348403571, −7.44067228883198094971892629392, −6.37830866342522138734970924697, −4.72703462418367686547485485489, −3.99180465655697594983669343182, −3.34215854617581809646860925291, −2.43461148908778162522817277956,
0.61242307618679237428340519287, 1.93118763869972678583781052199, 3.18014696673077928767826295427, 4.11061630430744772353274251190, 5.28028804399828649408982304974, 6.55049491136822664852327557367, 7.49233306649286331084051992976, 8.201573505940514491019095908614, 8.607955160211576293172240010529, 9.191137810993167215268290476351