| L(s) = 1 | + (−1.08 − 0.905i)2-s + (−1.75 + 2.09i)3-s + (0.360 + 1.96i)4-s + (−2.18 − 0.472i)5-s + (3.79 − 0.682i)6-s + (−1.59 − 2.75i)7-s + (1.38 − 2.46i)8-s + (−0.772 − 4.38i)9-s + (1.94 + 2.49i)10-s + (0.876 + 0.506i)11-s + (−4.74 − 2.69i)12-s + (−3.83 + 3.21i)13-s + (−0.766 + 4.43i)14-s + (4.82 − 3.73i)15-s + (−3.74 + 1.41i)16-s + (5.77 + 1.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.768 − 0.640i)2-s + (−1.01 + 1.20i)3-s + (0.180 + 0.983i)4-s + (−0.977 − 0.211i)5-s + (1.55 − 0.278i)6-s + (−0.601 − 1.04i)7-s + (0.491 − 0.870i)8-s + (−0.257 − 1.46i)9-s + (0.615 + 0.788i)10-s + (0.264 + 0.152i)11-s + (−1.36 − 0.778i)12-s + (−1.06 + 0.891i)13-s + (−0.204 + 1.18i)14-s + (1.24 − 0.965i)15-s + (−0.935 + 0.354i)16-s + (1.40 + 0.247i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.416719 - 0.0955573i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.416719 - 0.0955573i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.08 + 0.905i)T \) |
| 5 | \( 1 + (2.18 + 0.472i)T \) |
| 19 | \( 1 + (-0.956 + 4.25i)T \) |
| good | 3 | \( 1 + (1.75 - 2.09i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (1.59 + 2.75i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.876 - 0.506i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.83 - 3.21i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.77 - 1.01i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-5.67 - 2.06i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-2.84 + 0.501i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.591 - 1.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.25T + 37T^{2} \) |
| 41 | \( 1 + (-4.05 + 4.83i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.48 + 1.63i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.20 + 6.85i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-7.88 - 2.87i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.555 - 3.15i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (8.86 + 3.22i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-5.98 + 1.05i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.39 + 0.509i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-9.08 + 10.8i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.30 - 3.60i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (4.37 + 7.58i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.27 + 3.89i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (2.56 - 14.5i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.08581135467123810785451594894, −10.44257917821788842618197871543, −9.685703325353687923865614748488, −8.958784216983725481410676804115, −7.45696316032473934890071391858, −6.87567232892444257985217067402, −5.03036014462193411133163082742, −4.15143876443608014767909073475, −3.33409674594175381948237059399, −0.61677756627385454510198271474,
0.890045639743795326152703143985, 2.82527749843149894799429307846, 5.15884709894774308880493669570, 5.87767254090918998175897127281, 6.82497837575492342635587725165, 7.58621098457158435933869135830, 8.261792473176051486866886424418, 9.535881309852543053845906356332, 10.55733919190190065413746918098, 11.53316344277255727166612907701
