| L(s) = 1 | + (−1.39 − 2.05i)2-s + (0.0541 − 0.998i)3-s + (−1.53 + 3.85i)4-s + (−1.61 + 0.748i)5-s + (−2.12 + 1.27i)6-s + (−0.975 + 3.51i)7-s + (5.20 − 1.14i)8-s + (−0.994 − 0.108i)9-s + (3.78 + 2.27i)10-s + (−2.09 + 1.98i)11-s + (3.76 + 1.74i)12-s + (−5.08 + 0.553i)13-s + (8.57 − 2.88i)14-s + (0.659 + 1.65i)15-s + (−3.57 − 3.38i)16-s + (−1.42 − 5.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.984 − 1.45i)2-s + (0.0312 − 0.576i)3-s + (−0.768 + 1.92i)4-s + (−0.723 + 0.334i)5-s + (−0.867 + 0.521i)6-s + (−0.368 + 1.32i)7-s + (1.84 − 0.405i)8-s + (−0.331 − 0.0360i)9-s + (1.19 + 0.720i)10-s + (−0.632 + 0.598i)11-s + (1.08 + 0.503i)12-s + (−1.41 + 0.153i)13-s + (2.29 − 0.771i)14-s + (0.170 + 0.427i)15-s + (−0.894 − 0.847i)16-s + (−0.345 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.140055 + 0.0805258i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.140055 + 0.0805258i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0541 + 0.998i)T \) |
| 59 | \( 1 + (7.64 + 0.737i)T \) |
| good | 2 | \( 1 + (1.39 + 2.05i)T + (-0.740 + 1.85i)T^{2} \) |
| 5 | \( 1 + (1.61 - 0.748i)T + (3.23 - 3.81i)T^{2} \) |
| 7 | \( 1 + (0.975 - 3.51i)T + (-5.99 - 3.60i)T^{2} \) |
| 11 | \( 1 + (2.09 - 1.98i)T + (0.595 - 10.9i)T^{2} \) |
| 13 | \( 1 + (5.08 - 0.553i)T + (12.6 - 2.79i)T^{2} \) |
| 17 | \( 1 + (1.42 + 5.13i)T + (-14.5 + 8.76i)T^{2} \) |
| 19 | \( 1 + (-3.59 + 2.73i)T + (5.08 - 18.3i)T^{2} \) |
| 23 | \( 1 + (0.197 - 0.372i)T + (-12.9 - 19.0i)T^{2} \) |
| 29 | \( 1 + (4.40 - 6.49i)T + (-10.7 - 26.9i)T^{2} \) |
| 31 | \( 1 + (2.97 + 2.25i)T + (8.29 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-6.60 - 1.45i)T + (33.5 + 15.5i)T^{2} \) |
| 41 | \( 1 + (-3.63 - 6.85i)T + (-23.0 + 33.9i)T^{2} \) |
| 43 | \( 1 + (6.02 + 5.71i)T + (2.32 + 42.9i)T^{2} \) |
| 47 | \( 1 + (5.25 + 2.43i)T + (30.4 + 35.8i)T^{2} \) |
| 53 | \( 1 + (7.20 - 4.33i)T + (24.8 - 46.8i)T^{2} \) |
| 61 | \( 1 + (-4.56 - 6.72i)T + (-22.5 + 56.6i)T^{2} \) |
| 67 | \( 1 + (-3.44 + 0.757i)T + (60.8 - 28.1i)T^{2} \) |
| 71 | \( 1 + (-13.9 - 6.44i)T + (45.9 + 54.1i)T^{2} \) |
| 73 | \( 1 + (-9.42 + 3.17i)T + (58.1 - 44.1i)T^{2} \) |
| 79 | \( 1 + (0.101 + 1.86i)T + (-78.5 + 8.54i)T^{2} \) |
| 83 | \( 1 + (1.40 + 8.54i)T + (-78.6 + 26.5i)T^{2} \) |
| 89 | \( 1 + (2.39 - 3.52i)T + (-32.9 - 82.6i)T^{2} \) |
| 97 | \( 1 + (-12.8 - 4.32i)T + (77.2 + 58.7i)T^{2} \) |
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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
−12.42797817197939339630780935673, −11.75694897426617619612708510671, −11.16442300541145045251961169713, −9.651803873790240310944193745214, −9.248515256015772257944672822592, −7.902288897259619763693845522726, −7.14847142760926188934756160620, −5.07581193767412647405674495028, −3.07465997128870438315658454570, −2.28330547577589761239007501019,
0.19196648862275071662776778704, 3.89271427789999512720747448184, 5.16588698081903473254167493051, 6.43847816458353635222477904901, 7.80490355771595389888392554496, 7.949337796741161739342648569086, 9.482660747219984694355853455304, 10.11126158165809051287234332848, 11.08674202930201845283678445811, 12.69274183546033483472726818634
