| L(s) = 1 | + (1.39 + 0.221i)2-s + (0.786 + 1.54i)3-s + (1.90 + 0.618i)4-s + (−3.06 + 3.94i)5-s + (0.756 + 2.32i)6-s + (2.82 + 2.82i)7-s + (2.52 + 1.28i)8-s + (−1.76 + 2.42i)9-s + (−5.15 + 4.83i)10-s + (−3.47 + 2.52i)11-s + (0.541 + 3.42i)12-s + (8.74 − 1.38i)13-s + (3.31 + 4.56i)14-s + (−8.50 − 1.62i)15-s + (3.23 + 2.35i)16-s + (−1.04 + 2.05i)17-s + ⋯ |
| L(s) = 1 | + (0.698 + 0.110i)2-s + (0.262 + 0.514i)3-s + (0.475 + 0.154i)4-s + (−0.613 + 0.789i)5-s + (0.126 + 0.388i)6-s + (0.403 + 0.403i)7-s + (0.315 + 0.160i)8-s + (−0.195 + 0.269i)9-s + (−0.515 + 0.483i)10-s + (−0.316 + 0.229i)11-s + (0.0451 + 0.285i)12-s + (0.672 − 0.106i)13-s + (0.237 + 0.326i)14-s + (−0.567 − 0.108i)15-s + (0.202 + 0.146i)16-s + (−0.0615 + 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.71049 + 1.32504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71049 + 1.32504i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 - 0.221i)T \) |
| 3 | \( 1 + (-0.786 - 1.54i)T \) |
| 5 | \( 1 + (3.06 - 3.94i)T \) |
| good | 7 | \( 1 + (-2.82 - 2.82i)T + 49iT^{2} \) |
| 11 | \( 1 + (3.47 - 2.52i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (-8.74 + 1.38i)T + (160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (1.04 - 2.05i)T + (-169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-13.1 + 4.26i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-4.16 + 26.3i)T + (-503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-10.5 - 3.43i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (1.07 + 3.30i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (0.317 + 2.00i)T + (-1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (-6.99 - 5.08i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-42.8 + 42.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (40.5 - 20.6i)T + (1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (35.5 + 69.7i)T + (-1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (37.9 - 52.2i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-69.3 + 50.3i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (50.9 - 100. i)T + (-2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (27.2 - 83.7i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-15.9 + 100. i)T + (-5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-38.8 - 12.6i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (70.9 + 36.1i)T + (4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-65.5 - 90.2i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (115. - 58.8i)T + (5.53e3 - 7.61e3i)T^{2} \) |
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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
−13.07523326581535482568754708968, −11.91195109836509556357579535381, −11.08209972581292452139914576271, −10.22369399379737022982063565569, −8.686991659588028579189400408832, −7.67860796127076401158290154319, −6.45547774819315683051403677905, −5.08560923879790160407343082043, −3.85472583681295413239268239205, −2.62833093479123326322431243091,
1.29574557237880781186632226753, 3.33873972046605540622099627679, 4.61242580506606588751192859755, 5.85861960282431236211738014811, 7.36241939825956106907892451585, 8.139512265521717106580677288223, 9.380951030463429177342438584353, 10.95973492233183362760678213168, 11.72250363718279315571233030749, 12.68386379386474381832785368605
