L(s) = 1 | + (1.44 − 0.832i)2-s + (−1.00 + 0.579i)3-s + (0.385 − 0.667i)4-s + (−0.965 + 1.67i)6-s + 2.43i·7-s + 2.04i·8-s + (−0.827 + 1.43i)9-s − 5.75·11-s + 0.893i·12-s + (1.38 + 0.797i)13-s + (2.02 + 3.51i)14-s + (2.47 + 4.28i)16-s + (5.18 − 2.99i)17-s + 2.75i·18-s + (−0.149 + 4.35i)19-s + ⋯ |
L(s) = 1 | + (1.01 − 0.588i)2-s + (−0.579 + 0.334i)3-s + (0.192 − 0.333i)4-s + (−0.394 + 0.682i)6-s + 0.920i·7-s + 0.723i·8-s + (−0.275 + 0.477i)9-s − 1.73·11-s + 0.258i·12-s + (0.383 + 0.221i)13-s + (0.541 + 0.938i)14-s + (0.618 + 1.07i)16-s + (1.25 − 0.725i)17-s + 0.649i·18-s + (−0.0342 + 0.999i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23747 + 0.905780i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23747 + 0.905780i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (0.149 - 4.35i)T \) |
good | 2 | \( 1 + (-1.44 + 0.832i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.00 - 0.579i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 2.43iT - 7T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 + (-1.38 - 0.797i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.18 + 2.99i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.814 - 0.470i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.30 + 2.26i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 - 2.89iT - 37T^{2} \) |
| 41 | \( 1 + (-3.15 - 5.46i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.93 + 2.26i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.75 - 4.47i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.90 - 1.09i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.39 + 9.35i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.26 + 9.11i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.874 - 0.504i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.41 + 7.65i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.87 - 5.12i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.80 - 6.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.11iT - 83T^{2} \) |
| 89 | \( 1 + (5.55 - 9.62i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.51 - 2.02i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.26564029460919626667607522568, −10.67705720009803021583214302636, −9.705742958716202684209856883855, −8.348071532393690525530421428109, −7.71379909658651080353319269403, −5.74413519962908226403313324178, −5.52479852575821719579146480567, −4.59256409829868487738841468163, −3.19714593174234350466089308318, −2.29452645865666748987171706570,
0.72687419741158034102610185750, 3.10441713509357274178914457461, 4.19900408053525096513995912229, 5.44052054291702785379917234279, 5.80809079583343459141776595975, 7.07745286518349848232079047951, 7.57147479024496440577537918474, 8.950471114285112585412863031613, 10.34597329687538265135282292193, 10.70064220516777061723345256914