| L(s) = 1 | + (2.81 − 8.67i)2-s + (−4.23 − 15.0i)3-s + (−41.4 − 30.1i)4-s + (53.2 − 17.2i)5-s + (−142. − 5.50i)6-s + (−14.1 + 19.4i)7-s + (−142. + 103. i)8-s + (−207. + 127. i)9-s − 510. i·10-s + (240. + 321. i)11-s + (−276. + 749. i)12-s + (323. + 105. i)13-s + (128. + 177. i)14-s + (−485. − 725. i)15-s + (−11.3 − 34.9i)16-s + (−463. − 1.42e3i)17-s + ⋯ |
| L(s) = 1 | + (0.498 − 1.53i)2-s + (−0.271 − 0.962i)3-s + (−1.29 − 0.941i)4-s + (0.952 − 0.309i)5-s + (−1.61 − 0.0624i)6-s + (−0.108 + 0.149i)7-s + (−0.785 + 0.570i)8-s + (−0.852 + 0.523i)9-s − 1.61i·10-s + (0.599 + 0.800i)11-s + (−0.553 + 1.50i)12-s + (0.531 + 0.172i)13-s + (0.175 + 0.241i)14-s + (−0.556 − 0.832i)15-s + (−0.0111 − 0.0341i)16-s + (−0.389 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0461i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0441998 + 1.91580i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0441998 + 1.91580i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.23 + 15.0i)T \) |
| 11 | \( 1 + (-240. - 321. i)T \) |
| good | 2 | \( 1 + (-2.81 + 8.67i)T + (-25.8 - 18.8i)T^{2} \) |
| 5 | \( 1 + (-53.2 + 17.2i)T + (2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (14.1 - 19.4i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-323. - 105. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (463. + 1.42e3i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-167. - 230. i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + 4.34e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-6.12e3 - 4.45e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (937. - 2.88e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-1.01e4 - 7.37e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (7.65e3 - 5.56e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 8.56e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (7.90e3 + 1.08e4i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.30e4 - 4.23e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (2.41e4 - 3.33e4i)T + (-2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-4.35e4 + 1.41e4i)T + (6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 2.23e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-8.80e3 + 2.86e3i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-2.09e4 + 2.88e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (3.50e4 + 1.13e4i)T + (2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (2.00e3 + 6.18e3i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 - 4.56e3iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (2.55e4 - 7.87e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−14.28333571914924515478709107675, −13.46849427948279498019854766825, −12.51829010123676710111296518979, −11.67380661996927014009082837904, −10.29961990743970478235304205679, −8.978640930916359469823965830794, −6.63117534142031023895188010512, −4.89260781277051661851266252212, −2.52265433737137764704992201302, −1.19300158977394378054992379704,
3.91863340047155566581279488960, 5.66305769500903728658817042944, 6.36199982324287661885785341560, 8.374989991461414265417997941701, 9.723484451431681438773475069866, 11.19112908076280330070464537727, 13.34683585356040134294958640665, 14.17684167188621762721311911139, 15.22901065509223145393433034942, 16.13440106658505663555685724616
