| L(s) = 1 | + (−0.221 + 1.39i)2-s + (−2.40 + 1.22i)3-s + (−1.90 − 0.618i)4-s + (−1.59 + 4.73i)5-s + (−1.17 − 3.62i)6-s + (−3.03 + 3.03i)7-s + (1.28 − 2.52i)8-s + (−1.01 + 1.39i)9-s + (−6.26 − 3.28i)10-s + (15.9 − 11.6i)11-s + (5.32 − 0.843i)12-s + (2.40 + 15.1i)13-s + (−3.56 − 4.91i)14-s + (−1.95 − 13.3i)15-s + (3.23 + 2.35i)16-s + (18.6 + 9.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 + 0.698i)2-s + (−0.801 + 0.408i)3-s + (−0.475 − 0.154i)4-s + (−0.319 + 0.947i)5-s + (−0.196 − 0.604i)6-s + (−0.433 + 0.433i)7-s + (0.160 − 0.315i)8-s + (−0.112 + 0.155i)9-s + (−0.626 − 0.328i)10-s + (1.45 − 1.05i)11-s + (0.444 − 0.0703i)12-s + (0.184 + 1.16i)13-s + (−0.254 − 0.350i)14-s + (−0.130 − 0.889i)15-s + (0.202 + 0.146i)16-s + (1.09 + 0.557i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.235887 + 0.664274i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.235887 + 0.664274i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.221 - 1.39i)T \) |
| 5 | \( 1 + (1.59 - 4.73i)T \) |
| good | 3 | \( 1 + (2.40 - 1.22i)T + (5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (3.03 - 3.03i)T - 49iT^{2} \) |
| 11 | \( 1 + (-15.9 + 11.6i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (-2.40 - 15.1i)T + (-160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-18.6 - 9.48i)T + (169. + 233. i)T^{2} \) |
| 19 | \( 1 + (3.15 - 1.02i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (16.3 + 2.58i)T + (503. + 163. i)T^{2} \) |
| 29 | \( 1 + (6.49 + 2.11i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-7.69 - 23.6i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (16.6 - 2.63i)T + (1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (-29.2 - 21.2i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-19.9 - 19.9i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (12.1 + 23.8i)T + (-1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-86.7 + 44.1i)T + (1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-63.0 + 86.8i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-65.0 + 47.2i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (55.4 + 28.2i)T + (2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (30.4 - 93.6i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (50.7 + 8.03i)T + (5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-35.6 - 11.5i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (1.11 - 2.19i)T + (-4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-73.7 - 101. i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (47.3 + 92.9i)T + (-5.53e3 + 7.61e3i)T^{2} \) |
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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
−16.10494301323714589653521247685, −14.63172085838604954759591053920, −13.98176841744001462720418933634, −12.00598577268676910132353690682, −11.15961509152536346104642998860, −9.845370181245683668250883591685, −8.428932474388856662635797074808, −6.66027022422258887713626767701, −5.86365984040669055932130075750, −3.85259891495301111643346244014,
0.897829304873237536834996009248, 3.95402090322453076425028077168, 5.66633254583295591377101000551, 7.38993098565278380051428862045, 9.044783056351033354468269768118, 10.18313053970323430511583188201, 11.82078630376515223170524265089, 12.21800528504404686243230584525, 13.27271128466373109260383290008, 14.80651566881029960202055042938
