| 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
−11.65014225246987975812425003128, −10.96722909158874098681380544889, −9.725224757661778414387063803603, −9.129378324216157047303647925432, −8.494789290570742214941148476326, −7.03411006764183661797210107523, −5.23131797459702987478750009094, −4.76348297636585737802389456236, −2.40333543117810920509680794135, −0.39700985110992300344305229464,
1.71998320720690470309793818438, 4.09225601382527878726216741481, 5.98806697696612485267639493015, 6.76443705339054214034373539555, 7.81580294254637343755225086807, 8.078147796335137592424141148361, 9.999217966403710912526683013266, 10.47513714557404045235131809804, 11.24873126314447215114850199943, 12.53332379459819382542845175620
