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
−15.86684627195273149633254218682, −15.33547568815940932946249196016, −14.08659002104236556491415701425, −13.40684379812532915792160452510, −11.72161592731310756458401554419, −10.03534423906623338177279652106, −8.762562231189854946954219496688, −7.38828376816023090746034482620, −5.07457302899881575708901544925, −3.46509102829978740760701448655,
3.26974517043189472291427415641, 4.78025445443902479535841473093, 7.66322599270599920674199952732, 8.364171713280274496926239256761, 10.00850706701366814824703970888, 12.07855482842601962094535192777, 12.83472658781114521169462111132, 14.03581535705258299118344346897, 15.00579331371891175354484848379, 16.24659854070756665608604884071