L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.86 + 3.23i)5-s + (0.5 − 0.866i)7-s + 0.999·8-s − 3.73·10-s + (−2.09 + 3.63i)11-s + (−0.232 − 0.401i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 7·17-s − 2.73·19-s + (1.86 − 3.23i)20-s + (−2.09 − 3.63i)22-s + (3.09 + 5.36i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.834 + 1.44i)5-s + (0.188 − 0.327i)7-s + 0.353·8-s − 1.18·10-s + (−0.632 + 1.09i)11-s + (−0.0643 − 0.111i)13-s + (0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s − 1.69·17-s − 0.626·19-s + (0.417 − 0.722i)20-s + (−0.447 − 0.774i)22-s + (0.645 + 1.11i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.039901646\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.039901646\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.86 - 3.23i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.09 - 3.63i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.232 + 0.401i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 23 | \( 1 + (-3.09 - 5.36i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.23 + 7.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.09 - 1.90i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.66T + 37T^{2} \) |
| 41 | \( 1 + (-4.73 - 8.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.73 - 4.73i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.633 + 1.09i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + (3.09 + 5.36i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.96 + 8.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.63 - 2.83i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + (7.56 - 13.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.29 - 12.6i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 + (-1.46 + 2.53i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.980555885188247690705119816529, −9.655091166123665786562986218603, −8.468794544737508245266320009176, −7.53472373221044476680995743646, −6.80506775421695516142253494543, −6.37099562595878266180189287326, −5.22599904277464858287026612790, −4.27597932065107134121208381388, −2.77467198021771906339500641098, −1.90393965717513917812347915940,
0.48399585579180060737933102351, 1.81852840755495423695993278953, 2.75358899345313692942159518059, 4.29312548658099977542858581654, 5.02029252943026081262982644510, 5.85967622638866360560991853017, 6.92448964616507388267743805429, 8.368879279348836825090239254095, 8.812982384896826395626248746592, 9.068308103028797184722936546606