| L(s) = 1 | + (−1.24 − 0.675i)2-s + (1.66 − 0.467i)3-s + (1.08 + 1.67i)4-s + (−0.690 + 0.454i)5-s + (−2.38 − 0.545i)6-s + (1.79 + 1.69i)7-s + (−0.218 − 2.81i)8-s + (2.56 − 1.55i)9-s + (1.16 − 0.0980i)10-s + (2.65 + 5.29i)11-s + (2.59 + 2.29i)12-s + (−4.45 + 1.92i)13-s + (−1.08 − 3.31i)14-s + (−0.939 + 1.08i)15-s + (−1.63 + 3.65i)16-s + (−1.63 − 1.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.878 − 0.477i)2-s + (0.962 − 0.269i)3-s + (0.543 + 0.839i)4-s + (−0.308 + 0.203i)5-s + (−0.974 − 0.222i)6-s + (0.677 + 0.638i)7-s + (−0.0772 − 0.997i)8-s + (0.854 − 0.519i)9-s + (0.368 − 0.0309i)10-s + (0.801 + 1.59i)11-s + (0.750 + 0.661i)12-s + (−1.23 + 0.533i)13-s + (−0.289 − 0.884i)14-s + (−0.242 + 0.278i)15-s + (−0.408 + 0.912i)16-s + (−0.396 − 0.472i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35776 + 0.271388i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35776 + 0.271388i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.24 + 0.675i)T \) |
| 3 | \( 1 + (-1.66 + 0.467i)T \) |
| good | 5 | \( 1 + (0.690 - 0.454i)T + (1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (-1.79 - 1.69i)T + (0.407 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-2.65 - 5.29i)T + (-6.56 + 8.82i)T^{2} \) |
| 13 | \( 1 + (4.45 - 1.92i)T + (8.92 - 9.45i)T^{2} \) |
| 17 | \( 1 + (1.63 + 1.94i)T + (-2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-4.54 - 3.81i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (4.49 + 4.75i)T + (-1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.241 + 0.0282i)T + (28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (-1.82 - 7.69i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (-4.74 + 0.836i)T + (34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (3.29 + 2.45i)T + (11.7 + 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.248 + 4.26i)T + (-42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (0.190 + 0.0452i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + (-1.90 - 3.30i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.37 + 4.73i)T + (-35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (-13.5 + 4.06i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (7.86 + 0.918i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (-1.73 - 0.632i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.83 + 0.666i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-6.75 + 5.03i)T + (22.6 - 75.6i)T^{2} \) |
| 83 | \( 1 + (-6.58 + 4.90i)T + (23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (-4.03 - 11.0i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (8.15 + 5.36i)T + (38.4 + 89.0i)T^{2} \) |
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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
−10.24574364614597260224542424212, −9.640908905030254320580889487560, −8.994855371378530834653895181454, −8.054139557920012759978487181835, −7.32227552859783567107741676710, −6.75042683614116767794479440337, −4.78881900064836946005641217905, −3.75705877535625135089081439767, −2.40205860908509688831698504096, −1.72418736419922032511783291854,
0.954777507042798754166537033287, 2.51150963327951444885592078782, 3.87226973843639323116166130318, 4.99623235817416045186344794584, 6.20497889341787428958094770478, 7.46141020535315074334729291369, 7.921458765529918002841891560331, 8.639357313171888797240190966667, 9.547913862796444358818450283542, 10.16038149906101202100391426255
