| L(s) = 1 | − 1.81·3-s + 4.15i·7-s − 5.70·9-s + (−2.29 − 10.7i)11-s − 23.6i·13-s + 4.15i·17-s + 29.0i·19-s − 7.54i·21-s + 24.3·23-s + 26.7·27-s + 13.9i·29-s + 9.50·31-s + (4.17 + 19.5i)33-s − 46.8·37-s + 43.0i·39-s + ⋯ |
| L(s) = 1 | − 0.605·3-s + 0.593i·7-s − 0.633·9-s + (−0.208 − 0.977i)11-s − 1.82i·13-s + 0.244i·17-s + 1.52i·19-s − 0.359i·21-s + 1.05·23-s + 0.988·27-s + 0.481i·29-s + 0.306·31-s + (0.126 + 0.592i)33-s − 1.26·37-s + 1.10i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.208 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7031161246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7031161246\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (2.29 + 10.7i)T \) |
| good | 3 | \( 1 + 1.81T + 9T^{2} \) |
| 7 | \( 1 - 4.15iT - 49T^{2} \) |
| 13 | \( 1 + 23.6iT - 169T^{2} \) |
| 17 | \( 1 - 4.15iT - 289T^{2} \) |
| 19 | \( 1 - 29.0iT - 361T^{2} \) |
| 23 | \( 1 - 24.3T + 529T^{2} \) |
| 29 | \( 1 - 13.9iT - 841T^{2} \) |
| 31 | \( 1 - 9.50T + 961T^{2} \) |
| 37 | \( 1 + 46.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 58.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 62.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 18.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 62.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 60.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 72.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 62.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 113.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 126. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 70.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 87.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 26.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 105.T + 9.40e3T^{2} \) |
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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
−10.20741428980551354843437312155, −8.822191799395560966218037272444, −8.389575745693724807611034651138, −7.50197021786915099623834501354, −6.17899701789594541550340979399, −5.67669805230699276762170980903, −5.08681361768245601562062122754, −3.47355865235361267002215711504, −2.78814013671783471834791065027, −1.05625103333519691459332934891,
0.26842497543238601932190305789, 1.81062442622822854940944423921, 3.04882472527632971032618466555, 4.52961209845868065880455989112, 4.83701245049011604237275197506, 6.17406228867933004017658682681, 6.92650146081336132768541309077, 7.48297790875563602495843153542, 8.915479013283293312520605417275, 9.257161887082921325813656761706
