| L(s) = 1 | + 1.41i·2-s + (−0.865 + 2.87i)3-s − 2.00·4-s − 4.52i·5-s + (−4.06 − 1.22i)6-s + 8.90·7-s − 2.82i·8-s + (−7.50 − 4.97i)9-s + 6.39·10-s − 1.10i·11-s + (1.73 − 5.74i)12-s + 23.9·13-s + 12.5i·14-s + (12.9 + 3.91i)15-s + 4.00·16-s + 11.7i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.288 + 0.957i)3-s − 0.500·4-s − 0.904i·5-s + (−0.677 − 0.204i)6-s + 1.27·7-s − 0.353i·8-s + (−0.833 − 0.552i)9-s + 0.639·10-s − 0.100i·11-s + (0.144 − 0.478i)12-s + 1.84·13-s + 0.899i·14-s + (0.866 + 0.261i)15-s + 0.250·16-s + 0.692i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.36991 + 1.01783i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36991 + 1.01783i\) |
| \(L(2)\) |
|
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 | \( 1 + (0.865 - 2.87i)T \) |
| 59 | \( 1 + 7.68iT \) |
| good | 5 | \( 1 + 4.52iT - 25T^{2} \) |
| 7 | \( 1 - 8.90T + 49T^{2} \) |
| 11 | \( 1 + 1.10iT - 121T^{2} \) |
| 13 | \( 1 - 23.9T + 169T^{2} \) |
| 17 | \( 1 - 11.7iT - 289T^{2} \) |
| 19 | \( 1 - 5.12T + 361T^{2} \) |
| 23 | \( 1 + 14.0iT - 529T^{2} \) |
| 29 | \( 1 - 22.9iT - 841T^{2} \) |
| 31 | \( 1 + 29.7T + 961T^{2} \) |
| 37 | \( 1 - 16.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 58.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 28.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 44.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 39.7iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 45.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 42.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 30.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 110.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 97.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 13.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.50iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 129.T + 9.40e3T^{2} \) |
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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
−11.16187512422616782994033387518, −10.68969601772240229771209321877, −9.277624027556826082896030537959, −8.597573416134683765104695897706, −8.038092185082330479305392450756, −6.35762137445455788139722290401, −5.42975634864367394907178691713, −4.65022329455520180688245568340, −3.70983164744274843653745008651, −1.18229933120197329813361065998,
1.13188364772689975742567225801, 2.29646199578852179939030280485, 3.65909494311450303909787432361, 5.18673361543609143375994606457, 6.21455520451786905584314548277, 7.38084581640654680351252938023, 8.159830593416740204065399522798, 9.162958210168635705259265559395, 10.70478166330392247805126351981, 11.14167304434217260292968334097
