| L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.400 + 1.10i)3-s + (0.766 − 0.642i)4-s + (−0.0612 + 2.23i)5-s + 1.17i·6-s + (−3.81 + 0.672i)7-s + (0.500 − 0.866i)8-s + (1.24 + 1.04i)9-s + (0.706 + 2.12i)10-s + (−1.57 + 2.73i)11-s + (0.400 + 1.10i)12-s + (−1.40 + 1.17i)13-s + (−3.35 + 1.93i)14-s + (−2.43 − 0.962i)15-s + (0.173 − 0.984i)16-s + (4.46 + 3.74i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.231 + 0.635i)3-s + (0.383 − 0.321i)4-s + (−0.0273 + 0.999i)5-s + 0.477i·6-s + (−1.44 + 0.254i)7-s + (0.176 − 0.306i)8-s + (0.416 + 0.349i)9-s + (0.223 + 0.670i)10-s + (−0.476 + 0.825i)11-s + (0.115 + 0.317i)12-s + (−0.389 + 0.326i)13-s + (−0.896 + 0.517i)14-s + (−0.628 − 0.248i)15-s + (0.0434 − 0.246i)16-s + (1.08 + 0.908i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0114 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0114 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05382 + 1.06597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05382 + 1.06597i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 5 | \( 1 + (0.0612 - 2.23i)T \) |
| 37 | \( 1 + (-5.80 - 1.80i)T \) |
| good | 3 | \( 1 + (0.400 - 1.10i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (3.81 - 0.672i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (1.57 - 2.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.40 - 1.17i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.46 - 3.74i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-2.67 + 7.36i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-0.0262 - 0.0453i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.42 - 1.97i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.88iT - 31T^{2} \) |
| 41 | \( 1 + (-2.75 + 2.30i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 - 2.80T + 43T^{2} \) |
| 47 | \( 1 + (-9.42 + 5.44i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (12.8 + 2.25i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-12.1 - 2.13i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.0728 - 0.0867i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.55 + 0.803i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (12.4 + 4.52i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + 4.68iT - 73T^{2} \) |
| 79 | \( 1 + (5.72 - 1.01i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.100 + 0.119i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.80 - 0.493i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (8.58 + 14.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.61165074981510916074955034359, −10.55895791606754626560184173737, −10.06275504572508759933543610857, −9.358651578408508432118154178426, −7.47823217403235682275702168111, −6.79409199851062583561527158792, −5.73515336435690449647362545474, −4.61340779494552158850755080465, −3.44782085787453599639932572201, −2.49627356114473406586285284795,
0.864788193645494325088382525638, 3.00681161568215514543829956084, 4.09285637138162362504077117580, 5.59720265937827018291246671135, 6.09174734227405604921479696258, 7.36121467980058653793624977183, 8.020208487223055701108663922343, 9.497864621987214171185664444223, 10.06338008452345607328260310091, 11.61473490791800706957817914605
