| L(s) = 1 | + (−1.76 + 0.785i)2-s + (−1.27 − 1.17i)3-s + (1.15 − 1.28i)4-s + (−0.197 + 2.22i)5-s + (3.16 + 1.07i)6-s + (1.35 − 2.34i)7-s + (0.159 − 0.490i)8-s + (0.234 + 2.99i)9-s + (−1.40 − 4.08i)10-s + (−0.748 + 0.333i)11-s + (−2.98 + 0.274i)12-s + (−2.49 − 1.11i)13-s + (−0.546 + 5.19i)14-s + (2.87 − 2.60i)15-s + (0.466 + 4.43i)16-s + (−2.45 + 7.56i)17-s + ⋯ |
| L(s) = 1 | + (−1.24 + 0.555i)2-s + (−0.734 − 0.678i)3-s + (0.579 − 0.643i)4-s + (−0.0882 + 0.996i)5-s + (1.29 + 0.439i)6-s + (0.510 − 0.885i)7-s + (0.0563 − 0.173i)8-s + (0.0781 + 0.996i)9-s + (−0.443 − 1.29i)10-s + (−0.225 + 0.100i)11-s + (−0.862 + 0.0791i)12-s + (−0.692 − 0.308i)13-s + (−0.145 + 1.38i)14-s + (0.741 − 0.671i)15-s + (0.116 + 1.10i)16-s + (−0.595 + 1.83i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.117108 + 0.270350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.117108 + 0.270350i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.27 + 1.17i)T \) |
| 5 | \( 1 + (0.197 - 2.22i)T \) |
| good | 2 | \( 1 + (1.76 - 0.785i)T + (1.33 - 1.48i)T^{2} \) |
| 7 | \( 1 + (-1.35 + 2.34i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.748 - 0.333i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (2.49 + 1.11i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (2.45 - 7.56i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.21 - 6.82i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.215 - 2.05i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (5.45 + 1.16i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (-6.92 + 1.47i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (1.38 - 1.00i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.64 - 0.732i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (4.43 - 7.68i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.05 + 0.650i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-0.613 - 1.88i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.67 - 1.63i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-2.29 + 1.02i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (5.67 - 1.20i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (3.34 + 10.3i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-12.1 - 8.83i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-6.97 - 1.48i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-1.78 - 1.98i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (8.19 + 5.95i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (5.78 + 1.22i)T + (88.6 + 39.4i)T^{2} \) |
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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
−12.53332379459819382542845175620, −11.24873126314447215114850199943, −10.47513714557404045235131809804, −9.999217966403710912526683013266, −8.078147796335137592424141148361, −7.81580294254637343755225086807, −6.76443705339054214034373539555, −5.98806697696612485267639493015, −4.09225601382527878726216741481, −1.71998320720690470309793818438,
0.39700985110992300344305229464, 2.40333543117810920509680794135, 4.76348297636585737802389456236, 5.23131797459702987478750009094, 7.03411006764183661797210107523, 8.494789290570742214941148476326, 9.129378324216157047303647925432, 9.725224757661778414387063803603, 10.96722909158874098681380544889, 11.65014225246987975812425003128
