| L(s) = 1 | + 4.14i·3-s + 1.70·7-s − 8.18·9-s + (−3.98 − 10.2i)11-s + 16.6·13-s + 0.512·17-s + 19.5i·19-s + 7.07i·21-s + 11.2i·23-s + 3.35i·27-s + 48.1i·29-s + 5.40·31-s + (42.5 − 16.5i)33-s − 0.530i·37-s + 68.8i·39-s + ⋯ |
| L(s) = 1 | + 1.38i·3-s + 0.243·7-s − 0.909·9-s + (−0.362 − 0.931i)11-s + 1.27·13-s + 0.0301·17-s + 1.02i·19-s + 0.336i·21-s + 0.488i·23-s + 0.124i·27-s + 1.65i·29-s + 0.174·31-s + (1.28 − 0.501i)33-s − 0.0143i·37-s + 1.76i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.730088769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.730088769\) |
| \(L(2)\) |
|
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 | \( 1 + (3.98 + 10.2i)T \) |
| good | 3 | \( 1 - 4.14iT - 9T^{2} \) |
| 7 | \( 1 - 1.70T + 49T^{2} \) |
| 13 | \( 1 - 16.6T + 169T^{2} \) |
| 17 | \( 1 - 0.512T + 289T^{2} \) |
| 19 | \( 1 - 19.5iT - 361T^{2} \) |
| 23 | \( 1 - 11.2iT - 529T^{2} \) |
| 29 | \( 1 - 48.1iT - 841T^{2} \) |
| 31 | \( 1 - 5.40T + 961T^{2} \) |
| 37 | \( 1 + 0.530iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 28.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 3.65T + 1.84e3T^{2} \) |
| 47 | \( 1 - 3.58iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 51.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 41.1T + 3.48e3T^{2} \) |
| 61 | \( 1 - 42.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 73.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 13.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 107.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 15.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 16.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 140.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 97.0iT - 9.40e3T^{2} \) |
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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
−10.04647701888148583122583997713, −9.162829588584905217941569813950, −8.540969394976327007782910256494, −7.74058156518123542296921066786, −6.35818601923448081077593695377, −5.58278622039647072865060683139, −4.76826603708746492254178703576, −3.70601570259327370458344101575, −3.18741139054617658199980594519, −1.36545853203050558912360005411,
0.55858388440662821846140420229, 1.73588676340797953596803048626, 2.60597851708121045555910190011, 4.04645722948803382454734416925, 5.13297407696658801653200657009, 6.27829378777717704514510090660, 6.78141711937267959247608752317, 7.73349000334713373808302733477, 8.239893035454385212040688874756, 9.209181837266546368183551874139
