L(s) = 1 | + (−0.866 − 1.11i)2-s − 1.73·3-s + (−0.500 + 1.93i)4-s + (1.49 + 1.93i)6-s + (2.59 − 1.11i)8-s + 2.99·9-s + (0.866 − 3.35i)12-s + (−3.5 − 1.93i)16-s − 4.47i·17-s + (−2.59 − 3.35i)18-s − 7.74i·19-s + 3.46·23-s + (−4.50 + 1.93i)24-s − 5.19·27-s − 7.74i·31-s + (0.866 + 5.59i)32-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.790i)2-s − 1.00·3-s + (−0.250 + 0.968i)4-s + (0.612 + 0.790i)6-s + (0.918 − 0.395i)8-s + 0.999·9-s + (0.250 − 0.968i)12-s + (−0.875 − 0.484i)16-s − 1.08i·17-s + (−0.612 − 0.790i)18-s − 1.77i·19-s + 0.722·23-s + (−0.918 + 0.395i)24-s − 1.00·27-s − 1.39i·31-s + (0.153 + 0.988i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.370211 - 0.477940i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.370211 - 0.477940i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 1.11i)T \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 7.74iT - 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 7.74iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 4.47iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 7.74iT - 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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
−11.35544349582753295005645378254, −10.76457588172491725039543385366, −9.678375887896220290858155919061, −8.978381830941629571058825037299, −7.53199024700368932420842637473, −6.79613763137732242710813556925, −5.24910132319504212874055274480, −4.22710936322728793186216207473, −2.58145881607263761644060066635, −0.68921781166120861241284768793,
1.43369041309139557867186498191, 4.09953679252022700648221542972, 5.37952273885941959794617987601, 6.11648504470573418558988894707, 7.09165322969349987606342131903, 8.072956056963245475517567963300, 9.152166823641662559002539591854, 10.36345374602888581229197077363, 10.64950906621406584874261537628, 11.96891336777763925558923614733