| 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
−13.18627139572674333129268346487, −11.93153585151185118365249496572, −11.45441553125141917265169776777, −10.83134601005623560122027998542, −9.409334642563926639360089301413, −7.32999830655085922433030385587, −6.40412259719653495853280413139, −4.88664907011598555662987408970, −3.40330260665496049892259084761, −1.43774032283561506094242734336,
4.09747753986043781124314668689, 5.16828528477797737822702902131, 5.75645417678065653660120250000, 7.12239030390052826810825563162, 8.479459278637474110019448479860, 9.826807662228895196364768894305, 11.04027170588560263600722451110, 12.27660996186228626714257448825, 13.02443236032602960497481803168, 14.60950209850413439671137644290
