| L(s) = 1 | + (−0.453 − 0.891i)2-s + (1.61 − 0.631i)3-s + (−0.587 + 0.809i)4-s + (−1.55 − 1.60i)5-s + (−1.29 − 1.15i)6-s + (−2.97 − 2.97i)7-s + (0.987 + 0.156i)8-s + (2.20 − 2.03i)9-s + (−0.729 + 2.11i)10-s + (4.73 + 1.53i)11-s + (−0.437 + 1.67i)12-s + (1.57 + 0.801i)13-s + (−1.30 + 4.00i)14-s + (−3.51 − 1.61i)15-s + (−0.309 − 0.951i)16-s + (−0.223 + 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 − 0.630i)2-s + (0.931 − 0.364i)3-s + (−0.293 + 0.404i)4-s + (−0.694 − 0.719i)5-s + (−0.528 − 0.469i)6-s + (−1.12 − 1.12i)7-s + (0.349 + 0.0553i)8-s + (0.734 − 0.678i)9-s + (−0.230 + 0.668i)10-s + (1.42 + 0.464i)11-s + (−0.126 + 0.483i)12-s + (0.436 + 0.222i)13-s + (−0.347 + 1.07i)14-s + (−0.908 − 0.417i)15-s + (−0.0772 − 0.237i)16-s + (−0.0541 + 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.680719 - 0.830246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.680719 - 0.830246i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.453 + 0.891i)T \) |
| 3 | \( 1 + (-1.61 + 0.631i)T \) |
| 5 | \( 1 + (1.55 + 1.60i)T \) |
| good | 7 | \( 1 + (2.97 + 2.97i)T + 7iT^{2} \) |
| 11 | \( 1 + (-4.73 - 1.53i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.57 - 0.801i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.223 - 1.40i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.09 - 1.50i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.86 - 0.951i)T + (13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (-4.57 - 3.32i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (5.23 - 3.80i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.16 + 2.29i)T + (-21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (3.58 - 1.16i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-6.26 + 6.26i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.62 + 0.732i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.801 - 5.05i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (3.44 + 10.5i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (2.99 - 9.23i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (6.39 + 1.01i)T + (63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (-8.85 + 12.1i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.19 - 8.23i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-2.20 + 3.03i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (16.6 + 2.63i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-0.351 + 1.08i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.549 - 3.47i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−12.61080741721887874345065847615, −12.04634709924454199460448699085, −10.57182512257232141313841098437, −9.450631479799924101774526916217, −8.836909064229862798249218758042, −7.57980569903829115091037802722, −6.68599316782416347344785008298, −4.11122463420220628531114673997, −3.55084548697294285022980727189, −1.28010475897261021194536725427,
2.87963783483525982918810112578, 4.04158156158227975709269669558, 6.01824652501209474081980821493, 6.96312637930197362304348519486, 8.242642973794588737185554548376, 9.101258138891357767682648792668, 9.831157676858963130731534765471, 11.19697323652172800897012117827, 12.34418386054497086243494751116, 13.63333778468720322559342796590
