| L(s) = 1 | + (3.38 + 4.66i)2-s + (1.60 + 4.94i)3-s + (−5.32 + 16.4i)4-s + (4.81 + 3.50i)5-s + (−17.6 + 24.2i)6-s + (−13.8 − 4.51i)7-s + (−6.83 + 2.22i)8-s + (−21.8 + 15.8i)9-s + 34.3i·10-s + (7.57 − 120. i)11-s − 89.6·12-s + (62.7 + 86.3i)13-s + (−26.0 − 80.1i)14-s + (−9.56 + 29.4i)15-s + (189. + 137. i)16-s + (95.3 − 131. i)17-s + ⋯ |
| L(s) = 1 | + (0.847 + 1.16i)2-s + (0.178 + 0.549i)3-s + (−0.333 + 1.02i)4-s + (0.192 + 0.140i)5-s + (−0.489 + 0.673i)6-s + (−0.283 − 0.0921i)7-s + (−0.106 + 0.0347i)8-s + (−0.269 + 0.195i)9-s + 0.343i·10-s + (0.0626 − 0.998i)11-s − 0.622·12-s + (0.371 + 0.511i)13-s + (−0.132 − 0.408i)14-s + (−0.0424 + 0.130i)15-s + (0.741 + 0.538i)16-s + (0.329 − 0.453i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.30271 + 1.80606i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30271 + 1.80606i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.60 - 4.94i)T \) |
| 11 | \( 1 + (-7.57 + 120. i)T \) |
| good | 2 | \( 1 + (-3.38 - 4.66i)T + (-4.94 + 15.2i)T^{2} \) |
| 5 | \( 1 + (-4.81 - 3.50i)T + (193. + 594. i)T^{2} \) |
| 7 | \( 1 + (13.8 + 4.51i)T + (1.94e3 + 1.41e3i)T^{2} \) |
| 13 | \( 1 + (-62.7 - 86.3i)T + (-8.82e3 + 2.71e4i)T^{2} \) |
| 17 | \( 1 + (-95.3 + 131. i)T + (-2.58e4 - 7.94e4i)T^{2} \) |
| 19 | \( 1 + (-241. + 78.5i)T + (1.05e5 - 7.66e4i)T^{2} \) |
| 23 | \( 1 + 106.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (1.46e3 + 475. i)T + (5.72e5 + 4.15e5i)T^{2} \) |
| 31 | \( 1 + (49.9 - 36.3i)T + (2.85e5 - 8.78e5i)T^{2} \) |
| 37 | \( 1 + (475. - 1.46e3i)T + (-1.51e6 - 1.10e6i)T^{2} \) |
| 41 | \( 1 + (1.38e3 - 450. i)T + (2.28e6 - 1.66e6i)T^{2} \) |
| 43 | \( 1 - 2.46e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (1.00e3 + 3.10e3i)T + (-3.94e6 + 2.86e6i)T^{2} \) |
| 53 | \( 1 + (-3.02e3 + 2.20e3i)T + (2.43e6 - 7.50e6i)T^{2} \) |
| 59 | \( 1 + (-614. + 1.89e3i)T + (-9.80e6 - 7.12e6i)T^{2} \) |
| 61 | \( 1 + (2.01e3 - 2.77e3i)T + (-4.27e6 - 1.31e7i)T^{2} \) |
| 67 | \( 1 - 784.T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-6.27e3 - 4.56e3i)T + (7.85e6 + 2.41e7i)T^{2} \) |
| 73 | \( 1 + (-5.83e3 - 1.89e3i)T + (2.29e7 + 1.66e7i)T^{2} \) |
| 79 | \( 1 + (-4.14e3 - 5.70e3i)T + (-1.20e7 + 3.70e7i)T^{2} \) |
| 83 | \( 1 + (-2.56e3 + 3.53e3i)T + (-1.46e7 - 4.51e7i)T^{2} \) |
| 89 | \( 1 + 9.78e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (1.43e4 - 1.03e4i)T + (2.73e7 - 8.41e7i)T^{2} \) |
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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
−16.21219402001449873955794384540, −15.12838450068692124802416825053, −14.03630878987765817979713759888, −13.34109280918046725634356992160, −11.46251129044454719391279873259, −9.837275906174815976706854585752, −8.211229338249106521754912194631, −6.61129037385003053908069057624, −5.34007628319505920376919254699, −3.66976926293158648702103734463,
1.76728029912762586514239494600, 3.56335675881673378265291303497, 5.48647433180779264194735756994, 7.52873016422823311252332593792, 9.505311163835643096771015611030, 10.87866921449880034909339232004, 12.23473853465450782250317942433, 12.90380203198055251425763368021, 13.94493709516992107080414947018, 15.17363456158993441060655886391
