| L(s) = 1 | + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (2.37 + 0.866i)5-s + (−0.407 + 2.31i)7-s + (−0.500 + 0.866i)8-s + (1.26 + 2.19i)10-s + (−4.03 + 1.46i)11-s + (0.581 − 0.487i)13-s + (−1.79 + 1.50i)14-s + (−0.939 + 0.342i)16-s + (2.46 + 4.26i)17-s + (3.62 − 6.27i)19-s + (−0.439 + 2.49i)20-s + (−4.03 − 1.46i)22-s + (0.0530 + 0.300i)23-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (0.0868 + 0.492i)4-s + (1.06 + 0.387i)5-s + (−0.154 + 0.873i)7-s + (−0.176 + 0.306i)8-s + (0.400 + 0.693i)10-s + (−1.21 + 0.442i)11-s + (0.161 − 0.135i)13-s + (−0.480 + 0.403i)14-s + (−0.234 + 0.0855i)16-s + (0.596 + 1.03i)17-s + (0.831 − 1.44i)19-s + (−0.0983 + 0.557i)20-s + (−0.859 − 0.312i)22-s + (0.0110 + 0.0627i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.51425 + 1.42862i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51425 + 1.42862i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2.37 - 0.866i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.407 - 2.31i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (4.03 - 1.46i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.581 + 0.487i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.46 - 4.26i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.62 + 6.27i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0530 - 0.300i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.37 - 1.99i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.49 + 8.46i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.78 - 6.55i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.75 - 3.15i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.53 + 0.557i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.315 + 1.78i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 0.573T + 53T^{2} \) |
| 59 | \( 1 + (-5.14 - 1.87i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.91 + 10.8i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.0320 + 0.0269i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.10 + 3.64i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.54 + 9.60i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.64 + 4.74i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (5.22 + 4.38i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-3.96 + 6.86i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.08 + 1.12i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.22953924341595753375940806317, −10.19517039093478268092467973336, −9.486957612121808924930512341130, −8.388378495115767332937404480693, −7.44249924041014042264203283153, −6.30467731876373948887672998068, −5.64855313145609700387665278755, −4.81197499335288176617580873319, −3.10285244971307657964504524058, −2.21502832433767840842256060554,
1.16757220556976732924341537735, 2.65936901214423419823560169510, 3.82877195926309138050573523402, 5.24500266862941771277889578887, 5.67253162381043156366504560020, 6.99634784124352951027503633142, 8.006040088986395280820967771644, 9.278411172831357480459351034640, 10.12443134438214955721077173763, 10.54856356513053755553375419155
