| L(s) = 1 | + (1.05 + 1.83i)2-s + (−1.78 + 1.49i)3-s + (−1.24 + 2.15i)4-s + (−0.222 − 1.25i)5-s + (−4.63 − 1.68i)6-s + (0.247 + 1.40i)7-s − 1.03·8-s + (0.421 − 2.39i)9-s + (2.07 − 1.74i)10-s + (2.41 + 2.02i)11-s + (−1.00 − 5.71i)12-s + (−0.714 − 4.05i)13-s + (−2.31 + 1.94i)14-s + (2.28 + 1.91i)15-s + (1.39 + 2.40i)16-s + (−0.508 − 0.880i)17-s + ⋯ |
| L(s) = 1 | + (0.749 + 1.29i)2-s + (−1.03 + 0.864i)3-s + (−0.622 + 1.07i)4-s + (−0.0993 − 0.563i)5-s + (−1.89 − 0.689i)6-s + (0.0935 + 0.530i)7-s − 0.366·8-s + (0.140 − 0.796i)9-s + (0.656 − 0.550i)10-s + (0.727 + 0.610i)11-s + (−0.290 − 1.64i)12-s + (−0.198 − 1.12i)13-s + (−0.618 + 0.519i)14-s + (0.589 + 0.494i)15-s + (0.347 + 0.602i)16-s + (−0.123 − 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.386540 + 1.04438i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.386540 + 1.04438i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (8.96 + 5.34i)T \) |
| good | 2 | \( 1 + (-1.05 - 1.83i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.78 - 1.49i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (0.222 + 1.25i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.247 - 1.40i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.41 - 2.02i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.714 + 4.05i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.508 + 0.880i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.95 - 3.38i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.69 + 4.66i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (8.13 - 6.82i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.50 + 8.51i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.19 + 6.78i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + 0.845T + 41T^{2} \) |
| 43 | \( 1 + (3.41 + 5.90i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (8.66 - 3.15i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (0.934 + 5.30i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.79 - 5.70i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (4.57 + 1.66i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.40 - 7.96i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (4.98 - 8.62i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.00 - 5.03i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (1.46 - 8.32i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (3.39 - 2.84i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (3.01 + 1.09i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (1.81 + 10.3i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−14.60950209850413439671137644290, −13.02443236032602960497481803168, −12.27660996186228626714257448825, −11.04027170588560263600722451110, −9.826807662228895196364768894305, −8.479459278637474110019448479860, −7.12239030390052826810825563162, −5.75645417678065653660120250000, −5.16828528477797737822702902131, −4.09747753986043781124314668689,
1.43774032283561506094242734336, 3.40330260665496049892259084761, 4.88664907011598555662987408970, 6.40412259719653495853280413139, 7.32999830655085922433030385587, 9.409334642563926639360089301413, 10.83134601005623560122027998542, 11.45441553125141917265169776777, 11.93153585151185118365249496572, 13.18627139572674333129268346487
