| L(s) = 1 | + (1.49 − 1.33i)2-s + (2.63 − 1.43i)3-s + (0.457 − 3.97i)4-s + (1.71 + 9.75i)5-s + (2.02 − 5.64i)6-s + (4.26 + 3.57i)7-s + (−4.60 − 6.54i)8-s + (4.89 − 7.55i)9-s + (15.5 + 12.2i)10-s + (2.25 − 12.8i)11-s + (−4.48 − 11.1i)12-s + (−2.65 + 7.28i)13-s + (11.1 − 0.333i)14-s + (18.5 + 23.2i)15-s + (−15.5 − 3.63i)16-s + (−6.20 + 3.58i)17-s + ⋯ |
| L(s) = 1 | + (0.746 − 0.665i)2-s + (0.878 − 0.477i)3-s + (0.114 − 0.993i)4-s + (0.343 + 1.95i)5-s + (0.337 − 0.941i)6-s + (0.608 + 0.510i)7-s + (−0.575 − 0.817i)8-s + (0.543 − 0.839i)9-s + (1.55 + 1.22i)10-s + (0.205 − 1.16i)11-s + (−0.374 − 0.927i)12-s + (−0.204 + 0.560i)13-s + (0.794 − 0.0238i)14-s + (1.23 + 1.54i)15-s + (−0.973 − 0.227i)16-s + (−0.365 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.95598 - 1.15149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.95598 - 1.15149i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.49 + 1.33i)T \) |
| 3 | \( 1 + (-2.63 + 1.43i)T \) |
| good | 5 | \( 1 + (-1.71 - 9.75i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-4.26 - 3.57i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-2.25 + 12.8i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (2.65 - 7.28i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (6.20 - 3.58i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-12.6 - 7.29i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (28.7 + 34.2i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (20.5 - 7.46i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (4.24 - 3.56i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-38.3 + 22.1i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (10.1 - 27.9i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-35.2 - 6.22i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (35.4 - 42.2i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 4.60T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-1.03 - 5.84i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-15.9 + 18.9i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (17.6 - 48.5i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-4.25 + 2.45i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-12.4 + 21.6i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (23.6 - 8.62i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-48.7 + 17.7i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-67.3 - 38.9i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-16.0 + 90.8i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.83906747447343987667568358836, −11.18081908406198735800663213784, −10.24562686836326683286963592345, −9.230711901580571041861158822504, −7.86387581930473062962207248908, −6.60746635878070106301958878073, −5.95705336644842243193577619926, −3.92399983682320779161616410501, −2.87041774321275676365080721509, −2.00806593810424004382429307231,
1.89722460591100632956464305222, 3.96963589726905424932931129926, 4.72419635259609958702277849943, 5.50235222362076248120658682458, 7.50123922102335447161950463136, 8.045282367174255733426303222990, 9.153611956929017507486681069516, 9.810388612187966637481405358553, 11.61331585312746055968280981938, 12.55753706325212930050118257514
