| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 2.00·4-s − 2.23i·5-s + 2.44·6-s + 3.80i·7-s + 2.82·8-s + 2.99·9-s − 3.16i·10-s − 16.1i·11-s + 3.46·12-s + 12.9·13-s + 5.38i·14-s − 3.87i·15-s + 4.00·16-s − 10.8i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.500·4-s − 0.447i·5-s + 0.408·6-s + 0.543i·7-s + 0.353·8-s + 0.333·9-s − 0.316i·10-s − 1.47i·11-s + 0.288·12-s + 0.997·13-s + 0.384i·14-s − 0.258i·15-s + 0.250·16-s − 0.636i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.700423969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.700423969\) |
| \(L(2)\) |
|
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 | \( 1 - 1.73T \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-11.6 + 19.8i)T \) |
| good | 7 | \( 1 - 3.80iT - 49T^{2} \) |
| 11 | \( 1 + 16.1iT - 121T^{2} \) |
| 13 | \( 1 - 12.9T + 169T^{2} \) |
| 17 | \( 1 + 10.8iT - 289T^{2} \) |
| 19 | \( 1 - 17.1iT - 361T^{2} \) |
| 29 | \( 1 + 10.1T + 841T^{2} \) |
| 31 | \( 1 - 29.2T + 961T^{2} \) |
| 37 | \( 1 - 3.06iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 55.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 34.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 49.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 28.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 24.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 52.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 7.48iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 121.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 132.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 18.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 46.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 17.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 31.4iT - 9.40e3T^{2} \) |
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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
−10.28954451634396862396198644859, −9.083820926346191426148600965307, −8.517062479991851978285551416134, −7.70751014927082902347353964112, −6.32957757450938121727727600204, −5.75806903746031592102357096224, −4.59989633500824537227701222648, −3.53030115565785976927725634183, −2.66519565146318101237230263021, −1.12550915089572913122235056797,
1.53293046763256002240672693228, 2.75102996707702952499238634251, 3.85273562788989951059688149206, 4.57921765607657452726580530002, 5.86536954268201829642025550307, 6.96818183440341653458541507995, 7.41582714219712949022664045801, 8.551937318092949572154231441114, 9.603260030884524922403833464837, 10.40292615401875192795668425092
