| L(s) = 1 | − 1.41i·2-s + 1.99·4-s + 7·5-s + 7.07i·7-s − 8.48i·8-s − 9.89i·10-s + 16.9i·13-s + 10.0·14-s − 4.00·16-s + 4.24i·17-s − 16.9i·19-s + 13.9·20-s + 9·23-s + 24·25-s + 24·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.499·4-s + 1.40·5-s + 1.01i·7-s − 1.06i·8-s − 0.989i·10-s + 1.30i·13-s + 0.714·14-s − 0.250·16-s + 0.249i·17-s − 0.893i·19-s + 0.699·20-s + 0.391·23-s + 0.959·25-s + 0.923·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.172445814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.172445814\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.41iT - 4T^{2} \) |
| 5 | \( 1 - 7T + 25T^{2} \) |
| 7 | \( 1 - 7.07iT - 49T^{2} \) |
| 13 | \( 1 - 16.9iT - 169T^{2} \) |
| 17 | \( 1 - 4.24iT - 289T^{2} \) |
| 19 | \( 1 + 16.9iT - 361T^{2} \) |
| 23 | \( 1 - 9T + 529T^{2} \) |
| 29 | \( 1 + 22.6iT - 841T^{2} \) |
| 31 | \( 1 - 49T + 961T^{2} \) |
| 37 | \( 1 - 17T + 1.36e3T^{2} \) |
| 41 | \( 1 - 16.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 46.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 32T + 2.20e3T^{2} \) |
| 53 | \( 1 + 16T + 2.80e3T^{2} \) |
| 59 | \( 1 - 71T + 3.48e3T^{2} \) |
| 61 | \( 1 - 11.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 31T + 4.48e3T^{2} \) |
| 71 | \( 1 - 73T + 5.04e3T^{2} \) |
| 73 | \( 1 + 39.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 156. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 35.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 17T + 9.40e3T^{2} \) |
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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
−9.558296022145191707193031347545, −9.256209726012070296190809186602, −8.137259849319320112878156432762, −6.67810767473185864332906155709, −6.40052770563603359970333867415, −5.40518828495700970659082325800, −4.30410947364597596732599470767, −2.77278635843722614241562168013, −2.28336081677831405515085277785, −1.29623071465634502797093564536,
1.07775334221641359095138732183, 2.28305346052858889459469012778, 3.35441212460530370623727187718, 4.89536993201924350298564260728, 5.66315834758724107405792899010, 6.33578237492520727696628073616, 7.14840813063617745062015911594, 7.907600424657050321692870910694, 8.757519220548791085532378309870, 10.01524948606186479175692658378
