| L(s) = 1 | − 3.65·3-s − 9.09i·7-s + 4.33·9-s + (9.88 − 4.82i)11-s − 22.5i·13-s + 22.9i·17-s − 16.0i·19-s + 33.1i·21-s − 0.141·23-s + 17.0·27-s − 29.8i·29-s + 21.5·31-s + (−36.1 + 17.6i)33-s + 28.0·37-s + 82.1i·39-s + ⋯ |
| L(s) = 1 | − 1.21·3-s − 1.29i·7-s + 0.481·9-s + (0.898 − 0.438i)11-s − 1.73i·13-s + 1.34i·17-s − 0.842i·19-s + 1.58i·21-s − 0.00616·23-s + 0.631·27-s − 1.02i·29-s + 0.694·31-s + (−1.09 + 0.533i)33-s + 0.757·37-s + 2.10i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8306510489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8306510489\) |
| \(L(2)\) |
|
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 | \( 1 + (-9.88 + 4.82i)T \) |
| good | 3 | \( 1 + 3.65T + 9T^{2} \) |
| 7 | \( 1 + 9.09iT - 49T^{2} \) |
| 13 | \( 1 + 22.5iT - 169T^{2} \) |
| 17 | \( 1 - 22.9iT - 289T^{2} \) |
| 19 | \( 1 + 16.0iT - 361T^{2} \) |
| 23 | \( 1 + 0.141T + 529T^{2} \) |
| 29 | \( 1 + 29.8iT - 841T^{2} \) |
| 31 | \( 1 - 21.5T + 961T^{2} \) |
| 37 | \( 1 - 28.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 39.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 21.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 15.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 75.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 50.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 98.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 125.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 100.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 128. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 103. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 75.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 58.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 62.4T + 9.40e3T^{2} \) |
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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
−9.570560724629879579547705319909, −8.267868155390447490991708896215, −7.66725139059875751102881353713, −6.40560024678126414829681133284, −6.16393816385092693981828329859, −5.00377306699836693622101319923, −4.15331884324402361850775134110, −3.11314269492470116930415130141, −1.15952192175323513860038068970, −0.35940251098028265508182805948,
1.39285295731850412234256445834, 2.59424848498224001835739408204, 4.12884639741794093068206435063, 5.00057891561262027566725324741, 5.78558265251292105861245607367, 6.55662277897602916140632633939, 7.18499269985851944770077686766, 8.630930122653467220660715626254, 9.215131753776988525127584665196, 9.929142373480735196543023398805
