| L(s) = 1 | + (0.951 − 0.309i)2-s + (2.93 − 0.633i)3-s + (−2.42 + 1.76i)4-s + (−4.16 − 1.35i)5-s + (2.59 − 1.50i)6-s + (1.73 − 1.26i)7-s + (−4.11 + 5.66i)8-s + (8.19 − 3.71i)9-s − 4.38·10-s + (−9.06 + 6.23i)11-s + (−6 + 6.70i)12-s + (−2.70 − 8.33i)13-s + (1.26 − 1.73i)14-s + (−13.0 − 1.33i)15-s + (1.54 − 4.75i)16-s + (5.70 + 1.85i)17-s + ⋯ |
| L(s) = 1 | + (0.475 − 0.154i)2-s + (0.977 − 0.211i)3-s + (−0.606 + 0.440i)4-s + (−0.833 − 0.270i)5-s + (0.432 − 0.251i)6-s + (0.248 − 0.180i)7-s + (−0.514 + 0.707i)8-s + (0.910 − 0.412i)9-s − 0.438·10-s + (−0.823 + 0.566i)11-s + (−0.5 + 0.559i)12-s + (−0.208 − 0.641i)13-s + (0.0900 − 0.124i)14-s + (−0.871 − 0.0887i)15-s + (0.0965 − 0.297i)16-s + (0.335 + 0.109i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.28604 - 0.111346i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28604 - 0.111346i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.93 + 0.633i)T \) |
| 11 | \( 1 + (9.06 - 6.23i)T \) |
| good | 2 | \( 1 + (-0.951 + 0.309i)T + (3.23 - 2.35i)T^{2} \) |
| 5 | \( 1 + (4.16 + 1.35i)T + (20.2 + 14.6i)T^{2} \) |
| 7 | \( 1 + (-1.73 + 1.26i)T + (15.1 - 46.6i)T^{2} \) |
| 13 | \( 1 + (2.70 + 8.33i)T + (-136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (-5.70 - 1.85i)T + (233. + 169. i)T^{2} \) |
| 19 | \( 1 + (-25.1 - 18.2i)T + (111. + 343. i)T^{2} \) |
| 23 | \( 1 + 26.4iT - 529T^{2} \) |
| 29 | \( 1 + (-27.8 - 38.3i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (9.51 + 29.2i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (23.9 - 17.3i)T + (423. - 1.30e3i)T^{2} \) |
| 41 | \( 1 + (25.6 - 35.2i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + 39.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (2.52 - 3.47i)T + (-682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-10.7 + 3.48i)T + (2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (19.4 + 26.7i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-8.45 + 26.0i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 - 70.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + (83.8 + 27.2i)T + (4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (50.1 - 36.4i)T + (1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (19.4 + 59.9i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-104. - 34.0i)T + (5.57e3 + 4.04e3i)T^{2} \) |
| 89 | \( 1 - 9.66iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-30.9 - 95.2i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−16.24659854070756665608604884071, −15.00579331371891175354484848379, −14.03581535705258299118344346897, −12.83472658781114521169462111132, −12.07855482842601962094535192777, −10.00850706701366814824703970888, −8.364171713280274496926239256761, −7.66322599270599920674199952732, −4.78025445443902479535841473093, −3.26974517043189472291427415641,
3.46509102829978740760701448655, 5.07457302899881575708901544925, 7.38828376816023090746034482620, 8.762562231189854946954219496688, 10.03534423906623338177279652106, 11.72161592731310756458401554419, 13.40684379812532915792160452510, 14.08659002104236556491415701425, 15.33547568815940932946249196016, 15.86684627195273149633254218682
