L(s) = 1 | + (0.627 − 1.37i)2-s + (−0.841 − 0.970i)4-s + (0.142 − 0.989i)7-s + (−0.412 + 0.121i)8-s + (0.234 + 0.512i)11-s + (−1.27 − 0.817i)14-s + (0.0902 − 0.627i)16-s + 0.851·22-s + (−0.540 − 0.841i)23-s + (0.415 − 0.909i)25-s + (−1.08 + 0.694i)28-s + (−1.19 + 1.37i)29-s + (−1.16 − 0.750i)32-s + (0.239 + 0.153i)37-s + (1.61 + 0.474i)43-s + (0.300 − 0.658i)44-s + ⋯ |
L(s) = 1 | + (0.627 − 1.37i)2-s + (−0.841 − 0.970i)4-s + (0.142 − 0.989i)7-s + (−0.412 + 0.121i)8-s + (0.234 + 0.512i)11-s + (−1.27 − 0.817i)14-s + (0.0902 − 0.627i)16-s + 0.851·22-s + (−0.540 − 0.841i)23-s + (0.415 − 0.909i)25-s + (−1.08 + 0.694i)28-s + (−1.19 + 1.37i)29-s + (−1.16 − 0.750i)32-s + (0.239 + 0.153i)37-s + (1.61 + 0.474i)43-s + (0.300 − 0.658i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.523487113\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.523487113\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (0.540 + 0.841i)T \) |
good | 2 | \( 1 + (-0.627 + 1.37i)T + (-0.654 - 0.755i)T^{2} \) |
| 5 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 11 | \( 1 + (-0.234 - 0.512i)T + (-0.654 + 0.755i)T^{2} \) |
| 13 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 17 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 29 | \( 1 + (1.19 - 1.37i)T + (-0.142 - 0.989i)T^{2} \) |
| 31 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 37 | \( 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 41 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (-1.61 - 0.474i)T + (0.841 + 0.540i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.258 - 1.80i)T + (-0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 61 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \) |
| 71 | \( 1 + (-0.234 + 0.512i)T + (-0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 79 | \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 89 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 97 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
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
−9.733408889361485074091121283881, −8.908600313508296539871320997301, −7.71562680499792986404712912602, −7.04884466928912351187108309340, −5.92400737460783033258286232676, −4.69097840979598112966213372010, −4.25686149576791620424215546143, −3.31850846987658587647796765012, −2.26287029125150532460292332547, −1.14186110869076636067575625486,
1.97798315189398451564440292380, 3.41348371864380664062202379103, 4.34307036012119662106779134885, 5.47822049941964471406657989783, 5.74560835968242505181004130008, 6.63942931786595102275311015415, 7.56201960681698601211249245566, 8.158355746676953407098028025450, 9.009062472348048823234581371255, 9.670077040461811308013927926258