L(s) = 1 | + (2.35 + 0.764i)2-s + (−1.40 + 1.01i)3-s + (1.72 + 1.25i)4-s + (−0.789 − 2.42i)5-s + (−4.07 + 1.32i)6-s + (−0.100 + 0.137i)7-s + (−2.72 − 3.74i)8-s + (0.927 − 2.85i)9-s − 6.32i·10-s + (−7.69 + 7.85i)11-s − 3.68·12-s + (18.3 + 5.95i)13-s + (−0.341 + 0.247i)14-s + (3.57 + 2.59i)15-s + (−6.17 − 19.0i)16-s + (−19.3 + 6.27i)17-s + ⋯ |
L(s) = 1 | + (1.17 + 0.382i)2-s + (−0.467 + 0.339i)3-s + (0.430 + 0.312i)4-s + (−0.157 − 0.485i)5-s + (−0.679 + 0.220i)6-s + (−0.0143 + 0.0196i)7-s + (−0.340 − 0.468i)8-s + (0.103 − 0.317i)9-s − 0.632i·10-s + (−0.699 + 0.714i)11-s − 0.307·12-s + (1.41 + 0.458i)13-s + (−0.0243 + 0.0177i)14-s + (0.238 + 0.173i)15-s + (−0.385 − 1.18i)16-s + (−1.13 + 0.368i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.36681 + 0.319885i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36681 + 0.319885i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.40 - 1.01i)T \) |
| 11 | \( 1 + (7.69 - 7.85i)T \) |
good | 2 | \( 1 + (-2.35 - 0.764i)T + (3.23 + 2.35i)T^{2} \) |
| 5 | \( 1 + (0.789 + 2.42i)T + (-20.2 + 14.6i)T^{2} \) |
| 7 | \( 1 + (0.100 - 0.137i)T + (-15.1 - 46.6i)T^{2} \) |
| 13 | \( 1 + (-18.3 - 5.95i)T + (136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (19.3 - 6.27i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (-8.57 - 11.8i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 - 7.74T + 529T^{2} \) |
| 29 | \( 1 + (-22.4 + 30.9i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (13.0 - 40.2i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (41.7 + 30.3i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (27.1 + 37.4i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + 59.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (27.6 - 20.0i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (4.70 - 14.4i)T + (-2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-21.3 - 15.4i)T + (1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-60.2 + 19.5i)T + (3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + 2.91T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-29.9 - 92.0i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (10.1 - 14.0i)T + (-1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (50.1 + 16.3i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-22.3 + 7.25i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 - 97.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + (15.6 - 48.1i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−15.96622690739083376284521479537, −15.63084959372411453724996420114, −14.13821606790320908306014514887, −13.04208612747892918149305690746, −12.06519145380806212146468824659, −10.52696147664727676192524409334, −8.824750448146746232837587067801, −6.69954801816886492603407261794, −5.29323744001235587798667716057, −4.03731350739414275373144785718,
3.17871944348555400207830985126, 5.11956771808791459996793652974, 6.54746715551602522310265256778, 8.499761539939320743529410002410, 10.86877691553171002372304418116, 11.49034988539668826682906232951, 13.10800397928980713579001772101, 13.53686669228741378747727530807, 15.04422405786732899281577508653, 16.13922016415717773128141460423