| L(s) = 1 | + (0.651 + 0.816i)2-s + (0.202 − 0.885i)4-s + (0.0373 − 0.498i)5-s + (−1.65 − 2.86i)7-s + (2.73 − 1.31i)8-s + (0.431 − 0.294i)10-s + (−0.828 − 3.62i)11-s + (−3.87 − 2.64i)13-s + (1.26 − 3.21i)14-s + (1.22 + 0.589i)16-s + (0.343 + 4.57i)17-s + (4.44 + 4.12i)19-s + (−0.433 − 0.133i)20-s + (2.42 − 3.04i)22-s + (0.371 + 0.114i)23-s + ⋯ |
| L(s) = 1 | + (0.460 + 0.577i)2-s + (0.101 − 0.442i)4-s + (0.0167 − 0.222i)5-s + (−0.624 − 1.08i)7-s + (0.967 − 0.466i)8-s + (0.136 − 0.0930i)10-s + (−0.249 − 1.09i)11-s + (−1.07 − 0.732i)13-s + (0.337 − 0.859i)14-s + (0.305 + 0.147i)16-s + (0.0831 + 1.11i)17-s + (1.01 + 0.945i)19-s + (−0.0970 − 0.0299i)20-s + (0.517 − 0.648i)22-s + (0.0774 + 0.0239i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49022 - 0.647098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49022 - 0.647098i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (-6.26 - 1.92i)T \) |
| good | 2 | \( 1 + (-0.651 - 0.816i)T + (-0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.0373 + 0.498i)T + (-4.94 - 0.745i)T^{2} \) |
| 7 | \( 1 + (1.65 + 2.86i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.828 + 3.62i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (3.87 + 2.64i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.343 - 4.57i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (-4.44 - 4.12i)T + (1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-0.371 - 0.114i)T + (19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-1.29 + 3.29i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (-0.861 + 0.129i)T + (29.6 - 9.13i)T^{2} \) |
| 37 | \( 1 + (1.96 - 3.40i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.72 - 8.43i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-0.776 + 3.40i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (3.94 - 2.68i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (-6.05 - 2.91i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (5.18 + 0.782i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (7.73 + 7.17i)T + (5.00 + 66.8i)T^{2} \) |
| 71 | \( 1 + (0.859 - 0.265i)T + (58.6 - 39.9i)T^{2} \) |
| 73 | \( 1 + (0.798 + 0.544i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-0.500 - 0.867i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.24 - 3.17i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-0.672 - 1.71i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (0.213 + 0.936i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−10.96117131281131542135864912389, −10.30461723370486847278150310169, −9.622537498351731348911655935659, −8.104905050417031479111735218895, −7.37267811750902788925319897916, −6.28854525120057119837338348019, −5.53247375293284405918821088297, −4.39573825324205600222099778645, −3.18288091397667383746232403495, −0.985523902875408131351986822112,
2.34981860239585280899843650335, 2.93859761457674019711517970826, 4.51546754832940097915016630696, 5.31327387971230206622930874934, 6.98420929019411063187902810657, 7.40636949972946444926159031519, 8.986421854146851644576320295424, 9.559187484377172096705893189497, 10.71866904596047992910488231013, 11.76391778732876886532437631348
