| L(s) = 1 | + (1.22 + 0.707i)2-s + (2.54 + 1.58i)3-s + (0.999 + 1.73i)4-s + (0.790 + 0.456i)5-s + (1.99 + 3.74i)6-s + 2.82i·8-s + (3.98 + 8.07i)9-s + (0.645 + 1.11i)10-s + (12.6 − 7.27i)11-s + (−0.196 + 5.99i)12-s + 0.583·13-s + (1.29 + 2.41i)15-s + (−2.00 + 3.46i)16-s + (−18.6 + 10.7i)17-s + (−0.832 + 12.7i)18-s + (−8 + 13.8i)19-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.849 + 0.528i)3-s + (0.249 + 0.433i)4-s + (0.158 + 0.0913i)5-s + (0.333 + 0.623i)6-s + 0.353i·8-s + (0.442 + 0.896i)9-s + (0.0645 + 0.111i)10-s + (1.14 − 0.661i)11-s + (−0.0163 + 0.499i)12-s + 0.0448·13-s + (0.0861 + 0.161i)15-s + (−0.125 + 0.216i)16-s + (−1.09 + 0.633i)17-s + (−0.0462 + 0.705i)18-s + (−0.421 + 0.729i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.52659 + 1.85766i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.52659 + 1.85766i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (-2.54 - 1.58i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-0.790 - 0.456i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-12.6 + 7.27i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 0.583T + 169T^{2} \) |
| 17 | \( 1 + (18.6 - 10.7i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (8 - 13.8i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-33.6 - 19.4i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 35.7iT - 841T^{2} \) |
| 31 | \( 1 + (29.2 + 50.6i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (10 - 17.3i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 8.75iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 11.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-7.34 - 4.24i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-44.0 + 25.4i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-50.4 + 29.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-19.4 + 33.7i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (35.2 + 61.1i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-36.1 - 62.6i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (10.4 - 18.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 145. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (46.4 + 26.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 111.T + 9.40e3T^{2} \) |
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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
−11.69719902394466424447211068415, −10.90437678100174080370040075834, −9.700842552004208506407858974545, −8.822847381252210457567832794717, −7.989170523801049980286608650900, −6.75947581811643115345291466135, −5.74820298584236155327597510232, −4.30500516719907426906386759388, −3.59003693117319139788510754062, −2.10369322919487218932268070442,
1.40552595686447402707887855025, 2.67512947712054774424937063375, 3.92361830718743485591891101843, 5.06587292709086971939106101615, 6.79756924465124349189034960758, 7.03522069932577495039934018786, 8.932989168759032369789009995466, 9.111519195367766342466045758256, 10.55957399537955952760011930052, 11.55336105525568390272303213218
