| L(s) = 1 | + (−1.61 + 1.03i)2-s + (−0.142 + 0.989i)3-s + (0.698 − 1.53i)4-s + (−2.78 + 0.817i)5-s + (−0.797 − 1.74i)6-s + (−0.654 − 0.755i)7-s + (−0.0867 − 0.603i)8-s + (−0.959 − 0.281i)9-s + (3.64 − 4.21i)10-s + (−1.01 − 0.649i)11-s + (1.41 + 0.909i)12-s + (2.65 − 3.06i)13-s + (1.84 + 0.540i)14-s + (−0.413 − 2.87i)15-s + (2.96 + 3.42i)16-s + (−0.185 − 0.406i)17-s + ⋯ |
| L(s) = 1 | + (−1.14 + 0.733i)2-s + (−0.0821 + 0.571i)3-s + (0.349 − 0.765i)4-s + (−1.24 + 0.365i)5-s + (−0.325 − 0.712i)6-s + (−0.247 − 0.285i)7-s + (−0.0306 − 0.213i)8-s + (−0.319 − 0.0939i)9-s + (1.15 − 1.33i)10-s + (−0.304 − 0.195i)11-s + (0.408 + 0.262i)12-s + (0.736 − 0.849i)13-s + (0.492 + 0.144i)14-s + (−0.106 − 0.741i)15-s + (0.742 + 0.856i)16-s + (−0.0450 − 0.0985i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.419394 + 0.0519617i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.419394 + 0.0519617i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (-3.14 - 3.61i)T \) |
| good | 2 | \( 1 + (1.61 - 1.03i)T + (0.830 - 1.81i)T^{2} \) |
| 5 | \( 1 + (2.78 - 0.817i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (1.01 + 0.649i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.65 + 3.06i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.185 + 0.406i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.122 - 0.269i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.93 + 4.24i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.518 + 3.60i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-5.35 - 1.57i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-1.51 + 0.446i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.200 + 1.39i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 3.48T + 47T^{2} \) |
| 53 | \( 1 + (-7.57 - 8.74i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-8.82 + 10.1i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.602 + 4.19i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-3.01 + 1.93i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-10.0 + 6.43i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.65 + 10.2i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (4.68 - 5.40i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (0.0254 + 0.00747i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.50 + 10.4i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.0741 + 0.0217i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.87774376946884447207979130101, −9.975716421032724689946921919861, −9.153164788245167970753722890204, −8.089845464246880257150238912540, −7.73326337631267255082352515636, −6.69528931113023574110101326499, −5.62612726535147345094979732415, −4.07203990850645631820740086251, −3.26707116156937894435928945206, −0.49479389140658991699639951122,
1.02273516057200850542887295814, 2.48490505611563001940475140010, 3.85704617196842008699144438565, 5.22622465870034942151876758477, 6.68379289256730933732010902128, 7.61557320118212884974000383826, 8.536389072003218999731647361308, 8.909820629546726014515007914420, 10.08897159263462185988564816013, 11.14467094734131446178633268575
