| L(s) = 1 | − 1.41i·2-s + 1.73·3-s − 2.00·4-s − 2.44i·6-s + (1.07 − 6.91i)7-s + 2.82i·8-s + 2.99·9-s − 9.92·11-s − 3.46·12-s + 20.2·13-s + (−9.78 − 1.51i)14-s + 4.00·16-s − 18.9·17-s − 4.24i·18-s − 4.68i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.500·4-s − 0.408i·6-s + (0.152 − 0.988i)7-s + 0.353i·8-s + 0.333·9-s − 0.902·11-s − 0.288·12-s + 1.55·13-s + (−0.698 − 0.108i)14-s + 0.250·16-s − 1.11·17-s − 0.235i·18-s − 0.246i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.605853370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.605853370\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.07 + 6.91i)T \) |
| good | 11 | \( 1 + 9.92T + 121T^{2} \) |
| 13 | \( 1 - 20.2T + 169T^{2} \) |
| 17 | \( 1 + 18.9T + 289T^{2} \) |
| 19 | \( 1 + 4.68iT - 361T^{2} \) |
| 23 | \( 1 + 25.4iT - 529T^{2} \) |
| 29 | \( 1 + 15.9T + 841T^{2} \) |
| 31 | \( 1 + 20.9iT - 961T^{2} \) |
| 37 | \( 1 + 30.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 42.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 49.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.91T + 2.20e3T^{2} \) |
| 53 | \( 1 - 11.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 63.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 89.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 13.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 50.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + 78.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 69.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 68.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 156. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 154.T + 9.40e3T^{2} \) |
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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
−9.333791634097621297456765287522, −8.600971225039712514067603254209, −7.901644828908647054805176759233, −6.95479992291192477651271294042, −5.88687722135465949881071804092, −4.56227780434581516884383657892, −3.94840066768770440334521019829, −2.89618343112858503750986806321, −1.78687908887382415174368158294, −0.45314247599663475399566360895,
1.60726800457752929841556851100, 2.88737972463854829097321120197, 3.92644052898172438593777769091, 5.09090594225611979088005498386, 5.85537307125785052387499024408, 6.71718309758369085623468136853, 7.75774326639185013658352299280, 8.501486189031892721729805031921, 8.918213477219058230257614373845, 9.845054575211262878831731870309
