| L(s) = 1 | + (1 + i)2-s + 2i·4-s + (3.67 + 3.67i)7-s + (−2 + 2i)8-s + (−3.67 + 3.67i)13-s + 7.34i·14-s − 4·16-s + (6 + 6i)17-s − 7i·19-s + (−12 + 12i)23-s − 7.34·26-s + (−7.34 + 7.34i)28-s + 44.0i·29-s + 10·31-s + (−4 − 4i)32-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.5i)2-s + 0.5i·4-s + (0.524 + 0.524i)7-s + (−0.250 + 0.250i)8-s + (−0.282 + 0.282i)13-s + 0.524i·14-s − 0.250·16-s + (0.352 + 0.352i)17-s − 0.368i·19-s + (−0.521 + 0.521i)23-s − 0.282·26-s + (−0.262 + 0.262i)28-s + 1.52i·29-s + 0.322·31-s + (−0.125 − 0.125i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.025158095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.025158095\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-3.67 - 3.67i)T + 49iT^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 + (3.67 - 3.67i)T - 169iT^{2} \) |
| 17 | \( 1 + (-6 - 6i)T + 289iT^{2} \) |
| 19 | \( 1 + 7iT - 361T^{2} \) |
| 23 | \( 1 + (12 - 12i)T - 529iT^{2} \) |
| 29 | \( 1 - 44.0iT - 841T^{2} \) |
| 31 | \( 1 - 10T + 961T^{2} \) |
| 37 | \( 1 + (-25.7 - 25.7i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (29.3 - 29.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (18 + 18i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-30 + 30i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 44.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 23T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-33.0 - 33.0i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 132.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (33.0 - 33.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 103iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-42 + 42i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 44.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-91.8 - 91.8i)T + 9.40e3iT^{2} \) |
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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
−9.676493354320745593877381495420, −8.695489565754608250810091392742, −8.160527336359861328624171404509, −7.20410534184835704131720364697, −6.46549210422062251939581754552, −5.46025194465434557200640397596, −4.88682652067672352599154112521, −3.82509965056448842093734358976, −2.77829531734784283836527460925, −1.55117406178317701108927163732,
0.47235176191841275010984641252, 1.76729642506426286689958275448, 2.85505054058080536915117894341, 3.96091583284284710445232702053, 4.67881929546545423707907511143, 5.61513482656312201172586363771, 6.47508907852309275517782210862, 7.54546842403705408972621178987, 8.166405315480453239930628574964, 9.264980707515455872940524815795
