| L(s) = 1 | − 2-s − 3·4-s + 5.19i·5-s + 7·8-s − 5.19i·10-s + 11·11-s − 6.92i·13-s + 5·16-s + 24.2i·17-s − 3.46i·19-s − 15.5i·20-s − 11·22-s − 28·23-s − 2·25-s + 6.92i·26-s + ⋯ |
| L(s) = 1 | − 0.5·2-s − 0.750·4-s + 1.03i·5-s + 0.875·8-s − 0.519i·10-s + 11-s − 0.532i·13-s + 0.312·16-s + 1.42i·17-s − 0.182i·19-s − 0.779i·20-s − 0.5·22-s − 1.21·23-s − 0.0800·25-s + 0.266i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.234548 + 0.629110i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.234548 + 0.629110i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + 4T^{2} \) |
| 5 | \( 1 - 5.19iT - 25T^{2} \) |
| 11 | \( 1 - 11T + 121T^{2} \) |
| 13 | \( 1 + 6.92iT - 169T^{2} \) |
| 17 | \( 1 - 24.2iT - 289T^{2} \) |
| 19 | \( 1 + 3.46iT - 361T^{2} \) |
| 23 | \( 1 + 28T + 529T^{2} \) |
| 29 | \( 1 + 25T + 841T^{2} \) |
| 31 | \( 1 - 32.9iT - 961T^{2} \) |
| 37 | \( 1 + 58T + 1.36e3T^{2} \) |
| 41 | \( 1 - 3.46iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 26T + 1.84e3T^{2} \) |
| 47 | \( 1 - 76.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 31T + 2.80e3T^{2} \) |
| 59 | \( 1 + 8.66iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 13.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 52T + 4.48e3T^{2} \) |
| 71 | \( 1 + 64T + 5.04e3T^{2} \) |
| 73 | \( 1 + 6.92iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 17T + 6.24e3T^{2} \) |
| 83 | \( 1 - 53.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 79.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 91.7iT - 9.40e3T^{2} \) |
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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
−10.85280229300935185267603059167, −10.42378921133688084119529269865, −9.464250853767643929143422190990, −8.586855110305760800799446829070, −7.71917569020245977946208163348, −6.69127876400002716799517939535, −5.70906040619644591605787118543, −4.26144601823530679895157871110, −3.35181295582762685461679930357, −1.57833066933262737754955787161,
0.36270571936395091833929555732, 1.69762244042391011219510209170, 3.82138307530585473884559048388, 4.66023674674098844197714639843, 5.63029827797143237436178884623, 7.03599987511220942636917542216, 8.043712140877237405075307288687, 9.032802248052332866035318144247, 9.311972758558787273975536959208, 10.27937589761807149491095761633
