| L(s) = 1 | − 1.41·2-s + 2.00·4-s − 2.23i·5-s + 6.64i·7-s − 2.82·8-s + 3.16i·10-s + 9.54i·11-s + 10.3·13-s − 9.39i·14-s + 4.00·16-s − 15.5i·17-s − 0.664i·19-s − 4.47i·20-s − 13.4i·22-s + (−16.5 − 15.9i)23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s − 0.447i·5-s + 0.949i·7-s − 0.353·8-s + 0.316i·10-s + 0.867i·11-s + 0.794·13-s − 0.671i·14-s + 0.250·16-s − 0.917i·17-s − 0.0349i·19-s − 0.223i·20-s − 0.613i·22-s + (−0.719 − 0.694i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3517806430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3517806430\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (16.5 + 15.9i)T \) |
| good | 7 | \( 1 - 6.64iT - 49T^{2} \) |
| 11 | \( 1 - 9.54iT - 121T^{2} \) |
| 13 | \( 1 - 10.3T + 169T^{2} \) |
| 17 | \( 1 + 15.5iT - 289T^{2} \) |
| 19 | \( 1 + 0.664iT - 361T^{2} \) |
| 29 | \( 1 + 48.3T + 841T^{2} \) |
| 31 | \( 1 + 42.5T + 961T^{2} \) |
| 37 | \( 1 - 37.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 32.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 56.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 26.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 13.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 92.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 114. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 119. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 104.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 4.46T + 5.32e3T^{2} \) |
| 79 | \( 1 + 29.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 136. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 19.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 3.15iT - 9.40e3T^{2} \) |
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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
−8.718504033190138920909177937624, −8.014117021601784941897946269974, −7.24557560158524766500825203476, −6.31929995978479412887680578398, −5.55802370040798913040060860825, −4.69967118715309926319810801057, −3.55419227146056112757855227415, −2.37528146971079445143764993116, −1.57748839468491946109790893038, −0.11789663062888093183150730733,
1.12881542608890145254896753593, 2.18080296252456012233168899844, 3.70002563650108550780943053167, 3.82058585911760891455711495344, 5.66317482041818622074761671959, 6.01819142544590697828715958648, 7.26917296974243624616691865519, 7.48586586167673652981285053347, 8.574454756592468836309919517579, 9.075242942640449388913768166712
