L(s) = 1 | + i·2-s − 4-s + (1 − 2i)5-s − i·7-s − i·8-s + (2 + i)10-s + 5·11-s − 4i·13-s + 14-s + 16-s + 5·19-s + (−1 + 2i)20-s + 5i·22-s + 9i·23-s + (−3 − 4i)25-s + 4·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.447 − 0.894i)5-s − 0.377i·7-s − 0.353i·8-s + (0.632 + 0.316i)10-s + 1.50·11-s − 1.10i·13-s + 0.267·14-s + 0.250·16-s + 1.14·19-s + (−0.223 + 0.447i)20-s + 1.06i·22-s + 1.87i·23-s + (−0.600 − 0.800i)25-s + 0.784·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.047414182\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.047414182\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 9iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 13iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 + 9iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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
−8.892859740990070057605341090576, −7.81525480410426577040642305034, −7.38757921851535832777826796460, −6.39019997546870247413441295472, −5.59504673304903008351887190320, −5.13959725664855712548054563136, −4.02291739456649782699085467525, −3.38627285382185020667461616799, −1.66827752237531685909395942898, −0.75867701375121566635650294198,
1.25650241880376665435660990080, 2.20270623538410678589195443290, 3.08728682392209499772947350703, 3.95894335343993500232205444459, 4.77590708507078586646068419023, 5.89317436261654463177875560063, 6.61190049229403080180614047753, 7.11977385300912427656373129545, 8.393515000752099888774058015152, 9.002723344610718315109962323414