L(s) = 1 | + (2.5 − 0.866i)7-s − 5.19i·13-s + (0.5 − 0.866i)19-s + (−2.5 − 4.33i)25-s + (5.5 + 9.52i)31-s + (5.5 − 9.52i)37-s − 1.73i·43-s + (5.5 − 4.33i)49-s + (−6 − 3.46i)61-s + (13.5 − 7.79i)67-s + (1.5 − 0.866i)73-s + (4.5 + 2.59i)79-s + (−4.5 − 12.9i)91-s − 13.8i·97-s + (−6.5 + 11.2i)103-s + ⋯ |
L(s) = 1 | + (0.944 − 0.327i)7-s − 1.44i·13-s + (0.114 − 0.198i)19-s + (−0.5 − 0.866i)25-s + (0.987 + 1.71i)31-s + (0.904 − 1.56i)37-s − 0.264i·43-s + (0.785 − 0.618i)49-s + (−0.768 − 0.443i)61-s + (1.64 − 0.952i)67-s + (0.175 − 0.101i)73-s + (0.506 + 0.292i)79-s + (−0.471 − 1.36i)91-s − 1.40i·97-s + (−0.640 + 1.10i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.713671318\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.713671318\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-5.5 - 9.52i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 1.73iT - 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6 + 3.46i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-13.5 + 7.79i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.5 - 2.59i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−10.04369075740283898964419154714, −8.908570292508444541811797263013, −8.084055615272246010021346641484, −7.56052952340716718020469402504, −6.44295782931424713837663354124, −5.42368296416434588925194671934, −4.67850642741028613609107877559, −3.54829624376638419107881016540, −2.34767897603019312967976076449, −0.872390336948986254722597598955,
1.45580002141998516231787664156, 2.53215073632567230730026855483, 4.01589421443877902256444259515, 4.74671477932337098703228579638, 5.79249863017645001869900063218, 6.67589517465499948367453876473, 7.69646155047672918152114601450, 8.353888459261358224880990529514, 9.307985346581686852034792200507, 9.907146378929162580932039969213