L(s) = 1 | + (0.5 − 0.866i)2-s + (1.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.5 − 0.866i)6-s + (1 − 1.73i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (1.5 − 2.59i)11-s − 1.73i·12-s + (1 + 1.73i)13-s + (−0.999 − 1.73i)14-s + (−0.5 + 0.866i)16-s + 3·17-s + 3·18-s − 19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.866 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.612 − 0.353i)6-s + (0.377 − 0.654i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (0.452 − 0.783i)11-s − 0.499i·12-s + (0.277 + 0.480i)13-s + (−0.267 − 0.462i)14-s + (−0.125 + 0.216i)16-s + 0.727·17-s + 0.707·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.12303 - 0.772719i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12303 - 0.772719i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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
−10.80194123723058709284773998801, −10.30396656391235994292315939764, −9.205433117171909534296475886627, −8.509200536342683308378546150141, −7.48734328902989086705339282572, −6.18520155344595074273102980887, −4.81895662114354649792334659128, −3.96066438097556818285336863425, −3.04404321685483644591487605851, −1.52995577731522410467364358928,
1.82189865160928531626798301504, 3.21032342138418895131192584177, 4.34543626391456700795034458123, 5.65412790301850388454428958179, 6.56908728749290567611420359529, 7.73600669066641568157569878012, 8.124245844746655401012290506779, 9.274872603280131576111493044147, 9.910997074477443606702770574616, 11.56771461296296119974295062547