| L(s) = 1 | + (−0.930 + 3.47i)2-s + (0.635 + 2.93i)3-s + (−7.72 − 4.46i)4-s + (4.99 − 0.0933i)5-s + (−10.7 − 0.522i)6-s + (1.28 − 4.78i)7-s + (12.5 − 12.5i)8-s + (−8.19 + 3.72i)9-s + (−4.32 + 17.4i)10-s + (1.37 + 2.38i)11-s + (8.17 − 25.4i)12-s + (2.45 + 9.15i)13-s + (15.4 + 8.91i)14-s + (3.44 + 14.5i)15-s + (13.9 + 24.2i)16-s + (9.17 + 9.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.465 + 1.73i)2-s + (0.211 + 0.977i)3-s + (−1.93 − 1.11i)4-s + (0.999 − 0.0186i)5-s + (−1.79 − 0.0871i)6-s + (0.183 − 0.684i)7-s + (1.56 − 1.56i)8-s + (−0.910 + 0.413i)9-s + (−0.432 + 1.74i)10-s + (0.125 + 0.216i)11-s + (0.681 − 2.12i)12-s + (0.188 + 0.704i)13-s + (1.10 + 0.636i)14-s + (0.229 + 0.973i)15-s + (0.873 + 1.51i)16-s + (0.539 + 0.539i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.387i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.922 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.189714 + 0.942160i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.189714 + 0.942160i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.635 - 2.93i)T \) |
| 5 | \( 1 + (-4.99 + 0.0933i)T \) |
| good | 2 | \( 1 + (0.930 - 3.47i)T + (-3.46 - 2i)T^{2} \) |
| 7 | \( 1 + (-1.28 + 4.78i)T + (-42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (-1.37 - 2.38i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-2.45 - 9.15i)T + (-146. + 84.5i)T^{2} \) |
| 17 | \( 1 + (-9.17 - 9.17i)T + 289iT^{2} \) |
| 19 | \( 1 + 32.1iT - 361T^{2} \) |
| 23 | \( 1 + (-0.179 - 0.669i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + (30.1 - 17.3i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-12.8 + 22.2i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (13.0 + 13.0i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (-20.2 + 35.0i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-3.68 - 0.987i)T + (1.60e3 + 924.5i)T^{2} \) |
| 47 | \( 1 + (-0.993 + 3.70i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (31.2 - 31.2i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (36.8 + 21.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (6.56 + 11.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (44.9 - 12.0i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 114.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (18.7 - 18.7i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + (18.3 - 10.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (14.7 + 3.95i)T + (5.96e3 + 3.44e3i)T^{2} \) |
| 89 | \( 1 + 92.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (40.7 - 151. i)T + (-8.14e3 - 4.70e3i)T^{2} \) |
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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
−16.19876920474047400280755601752, −15.09505166398753635101696695134, −14.23117797410421331185866392552, −13.45298913999126335977732180089, −10.77363635775438638474262991616, −9.563550827535974385731050280418, −8.833015716425664853913264258075, −7.24199522237789920228847962013, −5.84048238952591852202690914910, −4.55005709774121974219479459233,
1.56612760689491072541305170817, 2.99959788779229181173619906094, 5.76218444708673548426325912356, 8.084846331789088377617857184449, 9.204572899561344656213314635462, 10.33527655676265720166405805481, 11.71581243122986323640248145409, 12.55297815586076251047686340227, 13.46583532297076947496484948925, 14.49263720901533209269968049614
