L(s) = 1 | − 2.10i·2-s − 1.90i·3-s − 2.43·4-s + (1.75 + 1.38i)5-s − 4.00·6-s + 0.116i·7-s + 0.913i·8-s − 0.615·9-s + (2.92 − 3.69i)10-s − 11-s + 4.62i·12-s − 2.04i·13-s + 0.246·14-s + (2.63 − 3.33i)15-s − 2.94·16-s − 6.68i·17-s + ⋯ |
L(s) = 1 | − 1.48i·2-s − 1.09i·3-s − 1.21·4-s + (0.783 + 0.620i)5-s − 1.63·6-s + 0.0441i·7-s + 0.322i·8-s − 0.205·9-s + (0.924 − 1.16i)10-s − 0.301·11-s + 1.33i·12-s − 0.567i·13-s + 0.0657·14-s + (0.681 − 0.860i)15-s − 0.736·16-s − 1.62i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.549319702\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.549319702\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.75 - 1.38i)T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + 2.10iT - 2T^{2} \) |
| 3 | \( 1 + 1.90iT - 3T^{2} \) |
| 7 | \( 1 - 0.116iT - 7T^{2} \) |
| 13 | \( 1 + 2.04iT - 13T^{2} \) |
| 17 | \( 1 + 6.68iT - 17T^{2} \) |
| 23 | \( 1 + 0.393iT - 23T^{2} \) |
| 29 | \( 1 + 7.51T + 29T^{2} \) |
| 31 | \( 1 - 0.564T + 31T^{2} \) |
| 37 | \( 1 + 7.53iT - 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 43 | \( 1 + 2.69iT - 43T^{2} \) |
| 47 | \( 1 - 13.2iT - 47T^{2} \) |
| 53 | \( 1 + 10.7iT - 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 7.51T + 61T^{2} \) |
| 67 | \( 1 - 4.18iT - 67T^{2} \) |
| 71 | \( 1 - 1.58T + 71T^{2} \) |
| 73 | \( 1 + 0.494iT - 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 2.11iT - 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 10.3iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.670082441762555945553685645724, −8.933948804414579477899065837052, −7.57888768486290060055374335267, −7.04057158362707161398821521197, −6.02167566563128715468913517128, −4.97537758226077994930154021337, −3.55711827072259615804338939459, −2.55545439615531529291682998324, −1.97204766350480311551684938470, −0.68585100339320585390567118836,
1.91641822374129751858341839473, 3.79408300276939846175958784529, 4.62381543608843741901469030010, 5.37058905911035458048787823332, 6.07374469333326601225010825668, 6.92437582597120393079902217117, 8.030325823400093357872308162660, 8.750212147267396093222911084312, 9.350938443961441208013208628240, 10.18014590496560498456925690704