L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.654 − 0.755i)3-s + (0.415 − 0.909i)4-s + (−0.959 − 0.281i)6-s − 0.563i·7-s + (−0.142 − 0.989i)8-s + (−0.142 + 0.989i)9-s − 0.830·11-s + (−0.959 + 0.281i)12-s + (−0.304 − 0.474i)14-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)18-s − 1.97i·19-s + (−0.425 + 0.368i)21-s + (−0.698 + 0.449i)22-s + ⋯ |
L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.654 − 0.755i)3-s + (0.415 − 0.909i)4-s + (−0.959 − 0.281i)6-s − 0.563i·7-s + (−0.142 − 0.989i)8-s + (−0.142 + 0.989i)9-s − 0.830·11-s + (−0.959 + 0.281i)12-s + (−0.304 − 0.474i)14-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)18-s − 1.97i·19-s + (−0.425 + 0.368i)21-s + (−0.698 + 0.449i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.246580327\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.246580327\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 3 | \( 1 + (0.654 + 0.755i)T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + 0.563iT - T^{2} \) |
| 11 | \( 1 + 0.830T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.97iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 0.563iT - T^{2} \) |
| 31 | \( 1 - 1.08iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 0.284T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.30T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.51iT - T^{2} \) |
| 97 | \( 1 - 0.830T + T^{2} \) |
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
−9.111574395020783185049358657916, −7.965273954320971316860709705919, −7.13211985254516484418847843169, −6.62647133182427370300378294441, −5.62377454060395846042048766948, −5.01185791478808765445415106960, −4.18901539981076442664756016075, −2.91223632622808443979817457861, −2.07722669255879354796332111870, −0.71142446747013975391390430563,
2.15659483998477071812008552309, 3.37295572007839185449154365206, 4.07211317649829003086666662675, 5.01124189604859047739106667333, 5.80365273311669929055291981360, 6.04219994701079969276257205102, 7.26822386975117055471082751863, 8.038819135943094733431140174035, 8.795977302632933997219909822602, 9.827644319519802074081448802060