| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0924 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0924 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1599561081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1599561081\) |
| \(L(2)\) |
|
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
−9.651705186279386186363228754968, −9.010435824536135757431591130461, −7.78095668978864180078812155733, −7.06998145015067921296852952910, −5.86025391992504665362268195126, −4.77940590589228721269612550459, −4.49005054760263257534858574513, −3.24428654855027228127151404976, −2.22925753111697893040112486126, −0.04911357084017039333094382092,
1.24277768398452064314330470692, 2.39580333593720831246905157842, 3.35030340435539717148686259356, 4.83108576399472856426852828737, 5.82711030443862119413317927160, 6.60325781490826725440625430617, 7.37106916247601647687749843642, 8.043821848161978628362681166592, 8.805606149538988595732256607622, 9.900117158525764565283283749391
