| L(s) = 1 | − 1.41i·2-s + (−2.64 − 1.41i)3-s − 2.00·4-s + (−2.00 + 3.74i)6-s + 2.64·7-s + 2.82i·8-s + (5 + 7.48i)9-s − 14.5i·11-s + (5.29 + 2.82i)12-s + 0.583·13-s − 3.74i·14-s + 4.00·16-s + 21.5i·17-s + (10.5 − 7.07i)18-s − 16·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.881 − 0.471i)3-s − 0.500·4-s + (−0.333 + 0.623i)6-s + 0.377·7-s + 0.353i·8-s + (0.555 + 0.831i)9-s − 1.32i·11-s + (0.440 + 0.235i)12-s + 0.0448·13-s − 0.267i·14-s + 0.250·16-s + 1.26i·17-s + (0.587 − 0.392i)18-s − 0.842·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3217572564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3217572564\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (2.64 + 1.41i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 11 | \( 1 + 14.5iT - 121T^{2} \) |
| 13 | \( 1 - 0.583T + 169T^{2} \) |
| 17 | \( 1 - 21.5iT - 289T^{2} \) |
| 19 | \( 1 + 16T + 361T^{2} \) |
| 23 | \( 1 + 38.8iT - 529T^{2} \) |
| 29 | \( 1 - 35.7iT - 841T^{2} \) |
| 31 | \( 1 + 58.4T + 961T^{2} \) |
| 37 | \( 1 + 20T + 1.36e3T^{2} \) |
| 41 | \( 1 + 8.75iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 11.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 8.48iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 50.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 58.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 70.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 72.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 20.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 145. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 53.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 111.T + 9.40e3T^{2} \) |
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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
−10.38564508643636730425794080037, −8.843862075806122848785721182213, −8.452136531838969658716896919643, −7.34172403430839823269856144562, −6.26813292943912495949421684718, −5.63726830822751839991483583358, −4.60360194904582596936579241115, −3.62334800887389138336154612765, −2.22622858282203119784563700600, −1.14319323700120410157533723690,
0.12487072401314354568837489369, 1.82276463167230130071608666968, 3.65928936748217591796796224161, 4.61068612317137178935944234056, 5.23422458914180974777171578042, 6.09077567028726848887316995808, 7.16750511754861520218488626006, 7.52608738701974931807263156004, 8.901607997200312576491637640304, 9.603268967113260367336190705627
