L(s) = 1 | + (−1.92 + 0.534i)2-s + (3.42 − 2.05i)4-s − 11.9·7-s + (−5.51 + 5.79i)8-s + 14.5i·11-s − 22.4i·13-s + (23.0 − 6.39i)14-s + (7.52 − 14.1i)16-s − 12.6i·17-s − 8.76i·19-s + (−7.76 − 28.0i)22-s + 4.99·23-s + (12.0 + 43.3i)26-s + (−41.0 + 24.6i)28-s + 2.74·29-s + ⋯ |
L(s) = 1 | + (−0.963 + 0.267i)2-s + (0.857 − 0.514i)4-s − 1.71·7-s + (−0.688 + 0.724i)8-s + 1.32i·11-s − 1.72i·13-s + (1.64 − 0.456i)14-s + (0.470 − 0.882i)16-s − 0.746i·17-s − 0.461i·19-s + (−0.352 − 1.27i)22-s + 0.217·23-s + (0.461 + 1.66i)26-s + (−1.46 + 0.880i)28-s + 0.0947·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6655689234\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6655689234\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.92 - 0.534i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 11.9T + 49T^{2} \) |
| 11 | \( 1 - 14.5iT - 121T^{2} \) |
| 13 | \( 1 + 22.4iT - 169T^{2} \) |
| 17 | \( 1 + 12.6iT - 289T^{2} \) |
| 19 | \( 1 + 8.76iT - 361T^{2} \) |
| 23 | \( 1 - 4.99T + 529T^{2} \) |
| 29 | \( 1 - 2.74T + 841T^{2} \) |
| 31 | \( 1 - 16.3iT - 961T^{2} \) |
| 37 | \( 1 - 32.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 42.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 16.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 48.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 94.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 43.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 56.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 61.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 39.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 99.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 10.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 140.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 54.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 14.1iT - 9.40e3T^{2} \) |
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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
−10.07349728686419013289069770256, −9.340436062641552779248327450344, −8.463039060849122176956464731694, −7.35135548409631492522959256006, −6.90520806841129498049664868526, −5.95178559318122159200000456685, −4.99464371324705696077630825335, −3.29540368262717622955533922638, −2.52390334624215224811458205444, −0.76316794231924306580202958544,
0.43652926094087877008061153399, 2.00547218826817322558981284064, 3.26920862614729232274941500390, 3.90995048060849201302734250637, 5.87077220442647558530405665979, 6.47762618260105416594344064755, 7.16375971588207930268461730364, 8.420503682524867428205114377432, 9.002899167053115046670791870001, 9.715579057287493442551050145982