| L(s) = 1 | − 3.20i·3-s + 3.06·5-s − 12.3·7-s − 1.30·9-s + 5.53·11-s + 6.70i·13-s − 9.83i·15-s − 5.36·17-s + (−1.17 − 18.9i)19-s + 39.5i·21-s − 9.72·23-s − 15.6·25-s − 24.7i·27-s − 14.4i·29-s + 36.0i·31-s + ⋯ |
| L(s) = 1 | − 1.06i·3-s + 0.612·5-s − 1.75·7-s − 0.144·9-s + 0.503·11-s + 0.516i·13-s − 0.655i·15-s − 0.315·17-s + (−0.0618 − 0.998i)19-s + 1.88i·21-s − 0.422·23-s − 0.624·25-s − 0.915i·27-s − 0.497i·29-s + 1.16i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0618 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4141749914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4141749914\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.17 + 18.9i)T \) |
| good | 3 | \( 1 + 3.20iT - 9T^{2} \) |
| 5 | \( 1 - 3.06T + 25T^{2} \) |
| 7 | \( 1 + 12.3T + 49T^{2} \) |
| 11 | \( 1 - 5.53T + 121T^{2} \) |
| 13 | \( 1 - 6.70iT - 169T^{2} \) |
| 17 | \( 1 + 5.36T + 289T^{2} \) |
| 23 | \( 1 + 9.72T + 529T^{2} \) |
| 29 | \( 1 + 14.4iT - 841T^{2} \) |
| 31 | \( 1 - 36.0iT - 961T^{2} \) |
| 37 | \( 1 - 32.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 60.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 60.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 58.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 24.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 107. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 45.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 42.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 70.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 52.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 2.68iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 138.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 106. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 79.3iT - 9.40e3T^{2} \) |
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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
−9.679487562300630929173171160671, −9.087711528742401885441402881088, −8.047306823705009732683397522781, −6.88031511886241187881951326396, −6.60797348112578530912004414226, −5.99444313184917947640375167284, −4.60883983258683921380270008471, −3.38284480523926770418801435756, −2.40480304500729751711629030700, −1.28805684367483648191954125102,
0.12049005880654991553113841988, 2.02418170386205082457347930756, 3.46348079633043982461986594201, 3.78523005152924465846447853702, 5.08949297423557925821318362893, 6.01695892691109206977701408071, 6.56273885988050132851802007897, 7.69196916370908070894644147502, 8.927721987438942388736165550445, 9.521714032783235164810822387218
