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
−16.13922016415717773128141460423, −15.04422405786732899281577508653, −13.53686669228741378747727530807, −13.10800397928980713579001772101, −11.49034988539668826682906232951, −10.86877691553171002372304418116, −8.499761539939320743529410002410, −6.54746715551602522310265256778, −5.11956771808791459996793652974, −3.17871944348555400207830985126,
4.03731350739414275373144785718, 5.29323744001235587798667716057, 6.69954801816886492603407261794, 8.824750448146746232837587067801, 10.52696147664727676192524409334, 12.06519145380806212146468824659, 13.04208612747892918149305690746, 14.13821606790320908306014514887, 15.63084959372411453724996420114, 15.96622690739083376284521479537