| L(s) = 1 | + (1.94 + 0.427i)2-s + (−0.161 − 0.986i)3-s + (1.76 + 0.818i)4-s + (0.704 + 2.53i)5-s + (0.107 − 1.98i)6-s + (0.381 + 0.361i)7-s + (−0.0794 − 0.0604i)8-s + (−0.947 + 0.319i)9-s + (0.283 + 5.22i)10-s + (3.19 − 3.76i)11-s + (0.521 − 1.87i)12-s + (−4.57 − 1.53i)13-s + (0.586 + 0.865i)14-s + (2.38 − 1.10i)15-s + (−2.65 − 3.12i)16-s + (−2.16 + 2.05i)17-s + ⋯ |
| L(s) = 1 | + (1.37 + 0.302i)2-s + (−0.0934 − 0.569i)3-s + (0.884 + 0.409i)4-s + (0.315 + 1.13i)5-s + (0.0439 − 0.810i)6-s + (0.144 + 0.136i)7-s + (−0.0280 − 0.0213i)8-s + (−0.315 + 0.106i)9-s + (0.0895 + 1.65i)10-s + (0.964 − 1.13i)11-s + (0.150 − 0.542i)12-s + (−1.26 − 0.427i)13-s + (0.156 + 0.231i)14-s + (0.616 − 0.285i)15-s + (−0.663 − 0.780i)16-s + (−0.526 + 0.498i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.12624 + 0.258083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12624 + 0.258083i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.161 + 0.986i)T \) |
| 59 | \( 1 + (-7.22 + 2.59i)T \) |
| good | 2 | \( 1 + (-1.94 - 0.427i)T + (1.81 + 0.839i)T^{2} \) |
| 5 | \( 1 + (-0.704 - 2.53i)T + (-4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-0.381 - 0.361i)T + (0.378 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-3.19 + 3.76i)T + (-1.77 - 10.8i)T^{2} \) |
| 13 | \( 1 + (4.57 + 1.53i)T + (10.3 + 7.86i)T^{2} \) |
| 17 | \( 1 + (2.16 - 2.05i)T + (0.920 - 16.9i)T^{2} \) |
| 19 | \( 1 + (1.05 - 2.65i)T + (-13.7 - 13.0i)T^{2} \) |
| 23 | \( 1 + (2.64 + 0.287i)T + (22.4 + 4.94i)T^{2} \) |
| 29 | \( 1 + (-0.405 + 0.0892i)T + (26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (-1.91 - 4.81i)T + (-22.5 + 21.3i)T^{2} \) |
| 37 | \( 1 + (7.39 - 5.62i)T + (9.89 - 35.6i)T^{2} \) |
| 41 | \( 1 + (-9.60 + 1.04i)T + (40.0 - 8.81i)T^{2} \) |
| 43 | \( 1 + (-5.32 - 6.26i)T + (-6.95 + 42.4i)T^{2} \) |
| 47 | \( 1 + (-1.37 + 4.95i)T + (-40.2 - 24.2i)T^{2} \) |
| 53 | \( 1 + (-0.654 + 12.0i)T + (-52.6 - 5.73i)T^{2} \) |
| 61 | \( 1 + (-0.915 - 0.201i)T + (55.3 + 25.6i)T^{2} \) |
| 67 | \( 1 + (-0.381 - 0.289i)T + (17.9 + 64.5i)T^{2} \) |
| 71 | \( 1 + (0.308 - 1.11i)T + (-60.8 - 36.6i)T^{2} \) |
| 73 | \( 1 + (-7.40 - 10.9i)T + (-27.0 + 67.8i)T^{2} \) |
| 79 | \( 1 + (0.769 - 4.69i)T + (-74.8 - 25.2i)T^{2} \) |
| 83 | \( 1 + (3.16 - 5.97i)T + (-46.5 - 68.6i)T^{2} \) |
| 89 | \( 1 + (9.64 - 2.12i)T + (80.7 - 37.3i)T^{2} \) |
| 97 | \( 1 + (-2.71 + 4.00i)T + (-35.9 - 90.1i)T^{2} \) |
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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
−12.83787132236079234878268691932, −12.02098408899050135092669700397, −11.13663759723359586296055545560, −9.940751558577558498661159831704, −8.403848304402703692147697054920, −6.94580299539322860418994197042, −6.35964380128050013895899906737, −5.36237238100839100071841829178, −3.76838941269949265762410747578, −2.56494291995041014442869346614,
2.28355746681320878923309451886, 4.31810889717081736071184410705, 4.59186026461253911076037200407, 5.75976455538450162378594460942, 7.17631124425003166538925966882, 9.005313970624758156280566110624, 9.536842423877017724457368338522, 11.01045595789676363811894680724, 12.20658750374430731585418072501, 12.39840228252273821392798371886
