L(s) = 1 | + (−1.05 − 1.69i)2-s − 3.44·3-s + (−1.77 + 3.58i)4-s + 4.88i·5-s + (3.62 + 5.84i)6-s + 2.64i·7-s + (7.96 − 0.763i)8-s + 2.84·9-s + (8.29 − 5.14i)10-s − 21.4·11-s + (6.10 − 12.3i)12-s + 13.0i·13-s + (4.49 − 2.79i)14-s − 16.8i·15-s + (−9.69 − 12.7i)16-s − 0.234·17-s + ⋯ |
L(s) = 1 | + (−0.527 − 0.849i)2-s − 1.14·3-s + (−0.443 + 0.896i)4-s + 0.976i·5-s + (0.604 + 0.974i)6-s + 0.377i·7-s + (0.995 − 0.0954i)8-s + 0.315·9-s + (0.829 − 0.514i)10-s − 1.95·11-s + (0.509 − 1.02i)12-s + 1.00i·13-s + (0.321 − 0.199i)14-s − 1.12i·15-s + (−0.606 − 0.795i)16-s − 0.0138·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0954 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0954 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.211528 + 0.232782i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.211528 + 0.232782i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.05 + 1.69i)T \) |
| 7 | \( 1 - 2.64iT \) |
good | 3 | \( 1 + 3.44T + 9T^{2} \) |
| 5 | \( 1 - 4.88iT - 25T^{2} \) |
| 11 | \( 1 + 21.4T + 121T^{2} \) |
| 13 | \( 1 - 13.0iT - 169T^{2} \) |
| 17 | \( 1 + 0.234T + 289T^{2} \) |
| 19 | \( 1 - 4.55T + 361T^{2} \) |
| 23 | \( 1 + 10.9iT - 529T^{2} \) |
| 29 | \( 1 + 34.6iT - 841T^{2} \) |
| 31 | \( 1 - 34.1iT - 961T^{2} \) |
| 37 | \( 1 - 54.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 37.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 4.84T + 1.84e3T^{2} \) |
| 47 | \( 1 - 72.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 21.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 34.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 63.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 18.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 47.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 55.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 95.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 71.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 159.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 90.4T + 9.40e3T^{2} \) |
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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
−15.61069407227110494045701580260, −13.94925853656422100644424624496, −12.66670340806617549049184863378, −11.57941419531719861037663892890, −10.82743976303860465335092982325, −9.970134196500561489795328847457, −8.203975876722358702457372932494, −6.70690368386254828997260112102, −5.00599829828753315537542256157, −2.72928896055146672295760503560,
0.40113159036813125091222350800, 5.06655906075711888793528675865, 5.61069342292341079188917965945, 7.40046103141546564497054509480, 8.517670852939903065223583235596, 10.13365786548915268893483858380, 10.94983618011028237571813328318, 12.62615706045204264173081583756, 13.47451582547582696682224941134, 15.20367196106219673576037097281