| L(s) = 1 | + (−1.40 + 0.144i)2-s + (−1.60 + 0.639i)3-s + (1.95 − 0.407i)4-s + (2.17 + 0.501i)5-s + (2.17 − 1.13i)6-s − 2.67i·7-s + (−2.69 + 0.855i)8-s + (2.18 − 2.05i)9-s + (−3.13 − 0.390i)10-s + (−1.85 + 1.34i)11-s + (−2.89 + 1.90i)12-s + (−2.84 − 2.06i)13-s + (0.387 + 3.76i)14-s + (−3.82 + 0.586i)15-s + (3.66 − 1.59i)16-s + (1.64 − 0.533i)17-s + ⋯ |
| L(s) = 1 | + (−0.994 + 0.102i)2-s + (−0.929 + 0.369i)3-s + (0.979 − 0.203i)4-s + (0.974 + 0.224i)5-s + (0.886 − 0.462i)6-s − 1.01i·7-s + (−0.953 + 0.302i)8-s + (0.727 − 0.686i)9-s + (−0.992 − 0.123i)10-s + (−0.558 + 0.405i)11-s + (−0.834 + 0.550i)12-s + (−0.789 − 0.573i)13-s + (0.103 + 1.00i)14-s + (−0.988 + 0.151i)15-s + (0.917 − 0.398i)16-s + (0.398 − 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.708845 - 0.133413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.708845 - 0.133413i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 - 0.144i)T \) |
| 3 | \( 1 + (1.60 - 0.639i)T \) |
| 5 | \( 1 + (-2.17 - 0.501i)T \) |
| good | 7 | \( 1 + 2.67iT - 7T^{2} \) |
| 11 | \( 1 + (1.85 - 1.34i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (2.84 + 2.06i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.64 + 0.533i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-6.93 + 2.25i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-6.63 + 4.81i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (5.87 + 1.90i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.24 + 0.406i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-7.58 - 5.50i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.57 + 6.30i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 2.46iT - 43T^{2} \) |
| 47 | \( 1 + (1.80 - 5.55i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.712 + 0.231i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (0.338 + 0.245i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (6.39 - 4.64i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.682 + 0.221i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (0.00682 - 0.0210i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.67 + 3.39i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (9.80 + 3.18i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.850 + 2.61i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (6.76 + 9.30i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.56 - 7.89i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.24747356792509011440254592961, −10.56495920973797595378660151086, −9.893175024464591399386694010125, −9.307539713524843865577086952392, −7.54102528589645881551028696197, −7.02323643070711656085846707816, −5.81408694778781061327467727513, −4.89843146360098986872768620708, −2.86737871258328656277642541805, −0.930940481457007172780569909868,
1.41134929085506370606248926019, 2.73416503190028627823281782349, 5.32706239205193237422352123380, 5.78168804207201041269731132657, 7.01426013514405745377194400910, 7.923864794686657456336208561793, 9.341631809075997475623985896308, 9.679760423682654067039826687840, 10.91357745555679959728628073251, 11.63481283963113339462301280315
