L(s) = 1 | + (−0.395 + 0.228i)2-s + (−1.39 − 2.41i)3-s + (−0.895 + 1.55i)4-s − 0.456i·5-s + (1.10 + 0.637i)6-s − 1.73i·8-s + (−2.39 + 4.14i)9-s + (0.104 + 0.180i)10-s + (−3.39 + 1.96i)11-s + 5·12-s + (3.5 + 0.866i)13-s + (−1.10 + 0.637i)15-s + (−1.39 − 2.41i)16-s + (−1.5 + 2.59i)17-s − 2.18i·18-s + (1.18 + 0.685i)19-s + ⋯ |
L(s) = 1 | + (−0.279 + 0.161i)2-s + (−0.805 − 1.39i)3-s + (−0.447 + 0.775i)4-s − 0.204i·5-s + (0.450 + 0.260i)6-s − 0.612i·8-s + (−0.798 + 1.38i)9-s + (0.0330 + 0.0571i)10-s + (−1.02 + 0.591i)11-s + 1.44·12-s + (0.970 + 0.240i)13-s + (−0.285 + 0.164i)15-s + (−0.348 − 0.604i)16-s + (−0.363 + 0.630i)17-s − 0.515i·18-s + (0.272 + 0.157i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.707051 + 0.0953921i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.707051 + 0.0953921i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-3.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.395 - 0.228i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.39 + 2.41i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + 0.456iT - 5T^{2} \) |
| 11 | \( 1 + (3.39 - 1.96i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.18 - 0.685i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.791 - 1.37i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.39 - 5.88i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 + (-6 + 3.46i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.79 + 3.92i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.68 - 8.11i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 9.57iT - 47T^{2} \) |
| 53 | \( 1 - 6.16T + 53T^{2} \) |
| 59 | \( 1 + (-10.6 - 6.15i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.37 + 12.7i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.87 - 2.23i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.79 - 2.18i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 3.46iT - 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 7.02iT - 83T^{2} \) |
| 89 | \( 1 + (13.9 - 8.07i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.31 + 3.64i)T + (48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.87955158866480827172024928370, −9.636019028007366712336264298018, −8.552258474671391122534829508611, −7.88306655228460058818132712041, −7.17463226117505783742998317534, −6.33124027713543121655735952926, −5.32543485852150628011804409859, −4.14286784632279039139331341837, −2.56374128699802295244760649674, −1.02454964244305663340257213510,
0.62688186587820710323029445047, 2.88501055851535715114662281431, 4.20362601693622709480390893399, 5.12697922980039934763406892369, 5.65447540937476669134007282665, 6.69056594169602440215232195658, 8.372282730158565522941767106115, 8.963191497352149379833444605616, 10.07538311460500720215384652670, 10.37533417338494648717995941067