L(s) = 1 | + (−0.186 + 0.982i)2-s + (−3.09 − 0.0725i)3-s + (−0.930 − 0.366i)4-s + (0.532 + 2.01i)5-s + (0.648 − 3.02i)6-s + (0.698 − 1.55i)7-s + (0.533 − 0.845i)8-s + (6.58 + 0.308i)9-s + (−2.08 + 0.146i)10-s + (1.67 − 1.88i)11-s + (2.85 + 1.20i)12-s + (0.229 + 3.25i)13-s + (1.39 + 0.976i)14-s + (−1.50 − 6.28i)15-s + (0.731 + 0.681i)16-s + (5.53 − 4.92i)17-s + ⋯ |
L(s) = 1 | + (−0.131 + 0.694i)2-s + (−1.78 − 0.0419i)3-s + (−0.465 − 0.183i)4-s + (0.238 + 0.902i)5-s + (0.264 − 1.23i)6-s + (0.264 − 0.588i)7-s + (0.188 − 0.299i)8-s + (2.19 + 0.102i)9-s + (−0.658 + 0.0463i)10-s + (0.504 − 0.567i)11-s + (0.823 + 0.346i)12-s + (0.0636 + 0.903i)13-s + (0.373 + 0.261i)14-s + (−0.387 − 1.62i)15-s + (0.182 + 0.170i)16-s + (1.34 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.318383 + 0.581518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.318383 + 0.581518i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.186 - 0.982i)T \) |
| 269 | \( 1 + (9.05 + 13.6i)T \) |
good | 3 | \( 1 + (3.09 + 0.0725i)T + (2.99 + 0.140i)T^{2} \) |
| 5 | \( 1 + (-0.532 - 2.01i)T + (-4.34 + 2.46i)T^{2} \) |
| 7 | \( 1 + (-0.698 + 1.55i)T + (-4.65 - 5.23i)T^{2} \) |
| 11 | \( 1 + (-1.67 + 1.88i)T + (-1.28 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.229 - 3.25i)T + (-12.8 + 1.82i)T^{2} \) |
| 17 | \( 1 + (-5.53 + 4.92i)T + (1.98 - 16.8i)T^{2} \) |
| 19 | \( 1 + (7.07 - 4.69i)T + (7.37 - 17.5i)T^{2} \) |
| 23 | \( 1 + (0.0778 - 3.32i)T + (-22.9 - 1.07i)T^{2} \) |
| 29 | \( 1 + (4.42 - 7.80i)T + (-14.8 - 24.8i)T^{2} \) |
| 31 | \( 1 + (-4.01 + 0.472i)T + (30.1 - 7.20i)T^{2} \) |
| 37 | \( 1 + (-3.75 - 6.99i)T + (-20.4 + 30.8i)T^{2} \) |
| 41 | \( 1 + (11.6 - 2.21i)T + (38.1 - 15.0i)T^{2} \) |
| 43 | \( 1 + (0.640 + 0.363i)T + (22.0 + 36.8i)T^{2} \) |
| 47 | \( 1 + (3.40 - 0.985i)T + (39.7 - 25.0i)T^{2} \) |
| 53 | \( 1 + (-4.31 + 5.88i)T + (-15.9 - 50.5i)T^{2} \) |
| 59 | \( 1 + (2.33 - 5.93i)T + (-43.1 - 40.2i)T^{2} \) |
| 61 | \( 1 + (-2.10 + 2.59i)T + (-12.7 - 59.6i)T^{2} \) |
| 67 | \( 1 + (-9.96 + 3.92i)T + (49.0 - 45.6i)T^{2} \) |
| 71 | \( 1 + (-6.58 - 9.43i)T + (-24.4 + 66.6i)T^{2} \) |
| 73 | \( 1 + (-2.41 - 6.57i)T + (-55.6 + 47.2i)T^{2} \) |
| 79 | \( 1 + (-0.584 + 0.450i)T + (20.1 - 76.3i)T^{2} \) |
| 83 | \( 1 + (0.441 - 4.69i)T + (-81.5 - 15.4i)T^{2} \) |
| 89 | \( 1 + (2.74 - 16.5i)T + (-84.2 - 28.6i)T^{2} \) |
| 97 | \( 1 + (4.18 + 5.17i)T + (-20.3 + 94.8i)T^{2} \) |
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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
−11.13340213234494690777130985610, −10.30591491559653025593026800306, −9.704903090921604881623964995006, −8.208205100074676997436963569099, −6.96187769513207615974903837591, −6.68503443825819349688587148018, −5.76760130399089200177141899281, −4.88297451721081420499018310513, −3.72852262203814387711427518556, −1.28221478396287476739429848637,
0.59445899620366981316766909742, 1.91532937770597666355720437359, 4.06423617923112397177386521179, 4.94572460359600685392500770451, 5.67592434762625255259832317902, 6.53429848817585709333371739080, 7.988346867721258449002781351328, 8.938597117019317595657123903446, 10.02472495408021199864458993685, 10.56881154880690972125204915066