| L(s) = 1 | − 1.09·3-s + (−2.20 − 0.370i)5-s + i·7-s − 1.80·9-s − 1.07i·11-s + 3.77·13-s + (2.41 + 0.405i)15-s + 5.08i·17-s − 0.279i·19-s − 1.09i·21-s + 4.13i·23-s + (4.72 + 1.63i)25-s + 5.25·27-s − 4.87i·29-s − 10.1·31-s + ⋯ |
| L(s) = 1 | − 0.631·3-s + (−0.986 − 0.165i)5-s + 0.377i·7-s − 0.601·9-s − 0.323i·11-s + 1.04·13-s + (0.622 + 0.104i)15-s + 1.23i·17-s − 0.0641i·19-s − 0.238i·21-s + 0.861i·23-s + (0.944 + 0.327i)25-s + 1.01·27-s − 0.904i·29-s − 1.81·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4021682784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4021682784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.20 + 0.370i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.09T + 3T^{2} \) |
| 11 | \( 1 + 1.07iT - 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 - 5.08iT - 17T^{2} \) |
| 19 | \( 1 + 0.279iT - 19T^{2} \) |
| 23 | \( 1 - 4.13iT - 23T^{2} \) |
| 29 | \( 1 + 4.87iT - 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 2.19T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 1.86T + 43T^{2} \) |
| 47 | \( 1 - 4.78iT - 47T^{2} \) |
| 53 | \( 1 + 5.30T + 53T^{2} \) |
| 59 | \( 1 + 5.22iT - 59T^{2} \) |
| 61 | \( 1 + 14.5iT - 61T^{2} \) |
| 67 | \( 1 + 9.74T + 67T^{2} \) |
| 71 | \( 1 + 4.18T + 71T^{2} \) |
| 73 | \( 1 + 12.4iT - 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 2.13T + 83T^{2} \) |
| 89 | \( 1 - 8.20T + 89T^{2} \) |
| 97 | \( 1 + 9.92iT - 97T^{2} \) |
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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
−8.694024617800389833496493051515, −8.086335517225459236352319448755, −7.34026154123118743114318496889, −6.10302057624814475252178224601, −5.88336014849798142836750208138, −4.80759151136498249356207295110, −3.83204969205300305408567107182, −3.16251047408332810403946915314, −1.61754191530431402367431944766, −0.18859525736777220094143783449,
1.00894446479607880647567666833, 2.69489889114973515068413205541, 3.61441355407949074959864991065, 4.46365838915303563632505825325, 5.31692780397071197423633826370, 6.16069626622983317589918202826, 7.08228709594752265347567375373, 7.52942847255012739926378449247, 8.641379281536053378950007557162, 9.018023360383853367194542263673
