| L(s) = 1 | + (−2.87 + 0.864i)3-s + 9.02·7-s + (7.50 − 4.96i)9-s − 21.8i·11-s − 21.6·13-s + 12.1i·17-s − 3.03·19-s + (−25.9 + 7.80i)21-s − 28.5i·23-s + (−17.2 + 20.7i)27-s + 12.0i·29-s − 2.19·31-s + (18.9 + 62.8i)33-s − 0.839·37-s + (62.2 − 18.7i)39-s + ⋯ |
| L(s) = 1 | + (−0.957 + 0.288i)3-s + 1.28·7-s + (0.833 − 0.551i)9-s − 1.98i·11-s − 1.66·13-s + 0.712i·17-s − 0.159·19-s + (−1.23 + 0.371i)21-s − 1.24i·23-s + (−0.639 + 0.768i)27-s + 0.415i·29-s − 0.0706·31-s + (0.573 + 1.90i)33-s − 0.0227·37-s + (1.59 − 0.480i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3598062690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3598062690\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.87 - 0.864i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 9.02T + 49T^{2} \) |
| 11 | \( 1 + 21.8iT - 121T^{2} \) |
| 13 | \( 1 + 21.6T + 169T^{2} \) |
| 17 | \( 1 - 12.1iT - 289T^{2} \) |
| 19 | \( 1 + 3.03T + 361T^{2} \) |
| 23 | \( 1 + 28.5iT - 529T^{2} \) |
| 29 | \( 1 - 12.0iT - 841T^{2} \) |
| 31 | \( 1 + 2.19T + 961T^{2} \) |
| 37 | \( 1 + 0.839T + 1.36e3T^{2} \) |
| 41 | \( 1 - 35.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 12.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 22.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 9.13iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 80.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 57.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 63.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 52.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 7.46T + 6.24e3T^{2} \) |
| 83 | \( 1 + 82.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 27.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 114.T + 9.40e3T^{2} \) |
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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
−9.154639117147347903108490781159, −8.350768607108572814223874725415, −7.59162346805034084942843589692, −6.50490624656426418420886319267, −5.71659708847113413487222056304, −4.93591016912510037457292692028, −4.22327847156081741558641159324, −2.86311113744821102096216861219, −1.37894235330521263508730252439, −0.12268138814292972874944522677,
1.57997211056449361047463984777, 2.30713014251499384853853600446, 4.30341134004149642817977400522, 4.91110782774890309121308057028, 5.41725722531728719401077746915, 6.87117959210374446224978108798, 7.38582466932400012762996295440, 7.902740114937053253405400953636, 9.415826005999651201510951551246, 9.917788842557357935805266813340
