| L(s) = 1 | + (−1.88 + 1.52i)2-s + (−1.53 + 0.804i)3-s + (0.811 − 3.81i)4-s + (−0.154 + 2.23i)5-s + (1.66 − 3.86i)6-s + (−3.12 − 0.838i)7-s + (2.09 + 4.11i)8-s + (1.70 − 2.46i)9-s + (−3.11 − 4.44i)10-s + (−3.96 − 0.416i)11-s + (1.82 + 6.50i)12-s + (0.956 − 1.18i)13-s + (7.19 − 3.20i)14-s + (−1.55 − 3.54i)15-s + (−3.12 − 1.39i)16-s + (4.14 − 2.11i)17-s + ⋯ |
| L(s) = 1 | + (−1.33 + 1.08i)2-s + (−0.885 + 0.464i)3-s + (0.405 − 1.90i)4-s + (−0.0691 + 0.997i)5-s + (0.680 − 1.57i)6-s + (−1.18 − 0.316i)7-s + (0.742 + 1.45i)8-s + (0.568 − 0.822i)9-s + (−0.986 − 1.40i)10-s + (−1.19 − 0.125i)11-s + (0.526 + 1.87i)12-s + (0.265 − 0.327i)13-s + (1.92 − 0.855i)14-s + (−0.401 − 0.915i)15-s + (−0.782 − 0.348i)16-s + (1.00 − 0.511i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.136006 - 0.0413174i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.136006 - 0.0413174i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.53 - 0.804i)T \) |
| 5 | \( 1 + (0.154 - 2.23i)T \) |
| good | 2 | \( 1 + (1.88 - 1.52i)T + (0.415 - 1.95i)T^{2} \) |
| 7 | \( 1 + (3.12 + 0.838i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (3.96 + 0.416i)T + (10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.956 + 1.18i)T + (-2.70 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-4.14 + 2.11i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.34 - 0.436i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.610 - 1.59i)T + (-17.0 - 15.3i)T^{2} \) |
| 29 | \( 1 + (-5.15 + 5.72i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (-0.568 - 0.631i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-1.50 + 9.48i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (11.0 - 1.16i)T + (40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-2.09 + 7.82i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (5.56 - 0.291i)T + (46.7 - 4.91i)T^{2} \) |
| 53 | \( 1 + (9.89 + 5.04i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (0.511 + 4.86i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (0.958 - 9.11i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-0.0988 - 0.00518i)T + (66.6 + 7.00i)T^{2} \) |
| 71 | \( 1 + (6.77 - 2.20i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.165 - 1.04i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (2.02 + 1.82i)T + (8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-7.09 + 10.9i)T + (-33.7 - 75.8i)T^{2} \) |
| 89 | \( 1 + (4.64 + 3.37i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.736 - 14.0i)T + (-96.4 + 10.1i)T^{2} \) |
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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
−11.80294424180567591633364239377, −10.53987839102647228999206028090, −10.18528726584473625221987499393, −9.480984666496535250684333144312, −7.950525602804128969242189576115, −7.10656437697035711879843520842, −6.26048484922400017880605129789, −5.44785935698651748778657630475, −3.33705624823361449054179995517, −0.21350158727012729868452823323,
1.37476794510245418185732140194, 3.06451570417243867972609517702, 5.00537132893284508154885155888, 6.34548169078203973252101138309, 7.75470738345689677439964186078, 8.516690934631855084602405231762, 9.746747582417436729720008836867, 10.23363218931199214240114742823, 11.33554847234118204982743111254, 12.34487865183725778576497327131
