| L(s) = 1 | + 3·3-s + (3.39 + 1.96i)5-s + (−6.39 − 11.0i)7-s + 9·9-s + (14.2 − 8.25i)11-s + (−1.39 + 2.42i)13-s + (10.1 + 5.88i)15-s − 2.54i·17-s − 21.5·19-s + (−19.1 − 33.2i)21-s + (2.60 + 1.50i)23-s + (−4.79 − 8.31i)25-s + 27·27-s + (13.1 − 7.61i)29-s + (13.6 − 23.5i)31-s + ⋯ |
| L(s) = 1 | + 3-s + (0.679 + 0.392i)5-s + (−0.914 − 1.58i)7-s + 9-s + (1.29 − 0.750i)11-s + (−0.107 + 0.186i)13-s + (0.679 + 0.392i)15-s − 0.149i·17-s − 1.13·19-s + (−0.914 − 1.58i)21-s + (0.113 + 0.0652i)23-s + (−0.191 − 0.332i)25-s + 27-s + (0.455 − 0.262i)29-s + (0.438 − 0.759i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.725998749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.725998749\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| good | 5 | \( 1 + (-3.39 - 1.96i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (6.39 + 11.0i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-14.2 + 8.25i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (1.39 - 2.42i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 2.54iT - 289T^{2} \) |
| 19 | \( 1 + 21.5T + 361T^{2} \) |
| 23 | \( 1 + (-2.60 - 1.50i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-13.1 + 7.61i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-13.6 + 23.5i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 10.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (34.5 + 19.9i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-17.0 - 29.6i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-58.1 + 33.6i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 100. iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (5.29 + 3.05i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-20.3 - 35.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-54.4 + 94.3i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 52.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 68.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-12.7 - 22.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (52.0 - 30.0i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 7.62iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-50.7 - 87.8i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.21880697850754836528760567275, −9.577323295584961252504241147029, −8.783300486697356567614131956908, −7.67448195319248945577004048149, −6.70494959450269473508175319731, −6.26884864496769884922748216417, −4.26454530284287362355828277931, −3.70131455162801110440201748004, −2.49581172823459236844651768699, −0.980422110614424418984232618323,
1.71051482389479678916026393753, 2.60027015351911080305266003717, 3.79448366997570974918677648936, 5.05398933174623130114866825982, 6.23273940440639986766501663639, 6.88813167569989772737104925355, 8.382555452373781966686207605886, 8.969464016709758297952077266338, 9.526199459332627856471536085291, 10.21404402830576027134386957199
