| L(s) = 1 | + (1.38 + 0.301i)2-s + (−0.540 + 0.841i)3-s + (1.81 + 0.833i)4-s + (−1.98 − 0.907i)5-s + (−1.00 + 0.999i)6-s + (4.66 + 1.36i)7-s + (2.26 + 1.69i)8-s + (−0.415 − 0.909i)9-s + (−2.47 − 1.85i)10-s + (−2.33 + 2.68i)11-s + (−1.68 + 1.07i)12-s + (1.24 − 0.365i)13-s + (6.03 + 3.29i)14-s + (1.83 − 1.18i)15-s + (2.61 + 3.03i)16-s + (−3.96 + 0.570i)17-s + ⋯ |
| L(s) = 1 | + (0.977 + 0.213i)2-s + (−0.312 + 0.485i)3-s + (0.909 + 0.416i)4-s + (−0.888 − 0.405i)5-s + (−0.408 + 0.407i)6-s + (1.76 + 0.517i)7-s + (0.799 + 0.600i)8-s + (−0.138 − 0.303i)9-s + (−0.781 − 0.586i)10-s + (−0.702 + 0.810i)11-s + (−0.486 + 0.311i)12-s + (0.345 − 0.101i)13-s + (1.61 + 0.881i)14-s + (0.474 − 0.305i)15-s + (0.652 + 0.757i)16-s + (−0.962 + 0.138i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.83282 + 0.890551i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83282 + 0.890551i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 - 0.301i)T \) |
| 3 | \( 1 + (0.540 - 0.841i)T \) |
| 23 | \( 1 + (-4.40 - 1.88i)T \) |
| good | 5 | \( 1 + (1.98 + 0.907i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-4.66 - 1.36i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (2.33 - 2.68i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.24 + 0.365i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (3.96 - 0.570i)T + (16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.618 + 4.30i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.820 + 5.70i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (3.68 + 5.72i)T + (-12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (6.57 - 3.00i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-5.12 + 11.2i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (5.98 + 3.84i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 1.24iT - 47T^{2} \) |
| 53 | \( 1 + (0.416 - 1.41i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-1.60 - 5.46i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (5.55 + 8.65i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-7.73 - 8.93i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (3.30 - 2.86i)T + (10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.0910 - 0.633i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (3.48 - 1.02i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (2.11 + 4.64i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (2.01 - 3.12i)T + (-36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-6.15 - 2.81i)T + (63.5 + 73.3i)T^{2} \) |
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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
−11.85373884354205339244338553867, −11.36580247873691853609400175864, −10.65593688520022427406948302486, −8.892157754624552467281192774837, −7.999069361848843097062611802061, −7.15286489630585965838909126921, −5.51391631870715553304609007443, −4.80530413687156931280750920290, −4.08156239356503477588761554570, −2.22372168308741001875937698070,
1.56021055944043364440639335969, 3.29654912376254315429235338997, 4.56725277538223726694533007394, 5.43307890873969450668061971743, 6.82801202836789264885788375713, 7.65436906973135925526255849471, 8.456571582078508596143065592260, 10.66488125916123575348696463833, 11.04451529652241210101174776476, 11.58853491595624655152315351874
