L(s) = 1 | + (0.516 − 1.31i)2-s − 1.28·3-s + (−1.46 − 1.36i)4-s + (−0.662 + 1.68i)6-s + (1.13 − 1.13i)7-s + (−2.54 + 1.22i)8-s − 1.35·9-s + (−2.32 − 2.32i)11-s + (1.87 + 1.74i)12-s − 1.36i·13-s + (−0.911 − 2.08i)14-s + (0.297 + 3.98i)16-s + (−5.25 + 5.25i)17-s + (−0.702 + 1.78i)18-s + (−3.69 − 3.69i)19-s + ⋯ |
L(s) = 1 | + (0.365 − 0.930i)2-s − 0.739·3-s + (−0.732 − 0.680i)4-s + (−0.270 + 0.688i)6-s + (0.430 − 0.430i)7-s + (−0.901 + 0.433i)8-s − 0.452·9-s + (−0.700 − 0.700i)11-s + (0.542 + 0.503i)12-s − 0.378i·13-s + (−0.243 − 0.558i)14-s + (0.0744 + 0.997i)16-s + (−1.27 + 1.27i)17-s + (−0.165 + 0.421i)18-s + (−0.848 − 0.848i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.134445 + 0.476462i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.134445 + 0.476462i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.516 + 1.31i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 1.28T + 3T^{2} \) |
| 7 | \( 1 + (-1.13 + 1.13i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.32 + 2.32i)T + 11iT^{2} \) |
| 13 | \( 1 + 1.36iT - 13T^{2} \) |
| 17 | \( 1 + (5.25 - 5.25i)T - 17iT^{2} \) |
| 19 | \( 1 + (3.69 + 3.69i)T + 19iT^{2} \) |
| 23 | \( 1 + (-0.911 - 0.911i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.37 - 2.37i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.242iT - 31T^{2} \) |
| 37 | \( 1 + 3.34iT - 37T^{2} \) |
| 41 | \( 1 + 2.66iT - 41T^{2} \) |
| 43 | \( 1 + 9.04iT - 43T^{2} \) |
| 47 | \( 1 + (7.87 + 7.87i)T + 47iT^{2} \) |
| 53 | \( 1 - 5.80T + 53T^{2} \) |
| 59 | \( 1 + (-5.91 + 5.91i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.67 + 6.67i)T + 61iT^{2} \) |
| 67 | \( 1 + 4.54iT - 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + (-1.49 + 1.49i)T - 73iT^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 3.26T + 83T^{2} \) |
| 89 | \( 1 - 9.77T + 89T^{2} \) |
| 97 | \( 1 + (-1.63 + 1.63i)T - 97iT^{2} \) |
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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
−10.91753619859117369823571284662, −10.40373713068341588294277886030, −8.928604144119519205837713489170, −8.273465309726422503399053778723, −6.63544444743841585455299456439, −5.64492723643771937971862297712, −4.82273362541843107169420061826, −3.63297784630383949777833783905, −2.19029148192097493019992065440, −0.29832917399720241200395437541,
2.60459308473230475507807198640, 4.42718921705749732584116080797, 5.10054280661950354765182904822, 6.09244807880241580698983125205, 6.92397918348966382487487524857, 8.017469817010241876652766460785, 8.842515301124894235289133681527, 9.876224906570016057325113423935, 11.17444698118621868883211059043, 11.81630498605627132285577783573