| L(s) = 1 | + (−1.11 − 1.23i)2-s + (0.560 + 1.63i)3-s + (−0.0801 + 0.762i)4-s + (2.09 + 0.772i)5-s + (1.40 − 2.51i)6-s + (0.694 − 1.20i)7-s + (−1.65 + 1.20i)8-s + (−2.37 + 1.83i)9-s + (−1.38 − 3.45i)10-s + (3.04 + 3.37i)11-s + (−1.29 + 0.295i)12-s + (0.454 − 0.504i)13-s + (−2.25 + 0.480i)14-s + (−0.0901 + 3.87i)15-s + (4.83 + 1.02i)16-s + (−1.73 + 1.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.787 − 0.874i)2-s + (0.323 + 0.946i)3-s + (−0.0400 + 0.381i)4-s + (0.938 + 0.345i)5-s + (0.572 − 1.02i)6-s + (0.262 − 0.454i)7-s + (−0.586 + 0.426i)8-s + (−0.790 + 0.612i)9-s + (−0.436 − 1.09i)10-s + (0.917 + 1.01i)11-s + (−0.373 + 0.0854i)12-s + (0.126 − 0.139i)13-s + (−0.603 + 0.128i)14-s + (−0.0232 + 0.999i)15-s + (1.20 + 0.257i)16-s + (−0.420 + 0.305i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06362 - 0.0396213i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06362 - 0.0396213i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.560 - 1.63i)T \) |
| 5 | \( 1 + (-2.09 - 0.772i)T \) |
| good | 2 | \( 1 + (1.11 + 1.23i)T + (-0.209 + 1.98i)T^{2} \) |
| 7 | \( 1 + (-0.694 + 1.20i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.04 - 3.37i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.454 + 0.504i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (1.73 - 1.26i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-5.56 + 4.04i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.104 + 0.0222i)T + (21.0 - 9.35i)T^{2} \) |
| 29 | \( 1 + (-1.87 + 0.835i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (5.90 + 2.63i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (0.886 + 2.72i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.63 + 4.03i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (0.613 - 1.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (10.8 - 4.81i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (1.07 + 0.781i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.15 - 1.28i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (2.52 + 2.80i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (12.9 + 5.78i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (10.9 + 7.96i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.83 - 5.63i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (8.20 - 3.65i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (1.89 + 18.0i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (-2.33 + 7.18i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-9.93 + 4.42i)T + (64.9 - 72.0i)T^{2} \) |
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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
−11.75585272128476646875668392077, −10.94562117448393796168648871461, −10.23490060427799608016932240749, −9.404666959150069524176156121584, −9.011963507725669630701207909397, −7.44094562923435918197445963975, −5.95028928598023251473211512612, −4.64922269159649391198220675067, −3.11875964556773197386963245577, −1.78369490217953471033692778065,
1.36408057334285772775836922613, 3.20046559048863684075973156211, 5.58763132336128930622096946501, 6.30913996172413456325103266826, 7.27231652721376039014368857214, 8.461216862981663460931732328462, 8.900693347009480736584991899617, 9.801248868996307749963759610683, 11.50145949559646294030506566649, 12.28089267878511627439942387615
