L(s) = 1 | + (2.98 − 9.17i)2-s + (21.8 − 15.8i)3-s + (28.3 + 20.5i)4-s + (33.3 + 102. i)5-s + (−80.4 − 247. i)6-s + (1.24e3 + 906. i)7-s + (1.27e3 − 923. i)8-s + (225. − 693. i)9-s + 1.04e3·10-s + (−3.08e3 + 3.15e3i)11-s + 944.·12-s + (1.30e3 − 4.02e3i)13-s + (1.20e4 − 8.73e3i)14-s + (2.35e3 + 1.71e3i)15-s + (−3.30e3 − 1.01e4i)16-s + (−5.18e3 − 1.59e4i)17-s + ⋯ |
L(s) = 1 | + (0.263 − 0.810i)2-s + (0.467 − 0.339i)3-s + (0.221 + 0.160i)4-s + (0.119 + 0.367i)5-s + (−0.152 − 0.468i)6-s + (1.37 + 0.998i)7-s + (0.878 − 0.638i)8-s + (0.103 − 0.317i)9-s + 0.329·10-s + (−0.699 + 0.714i)11-s + 0.157·12-s + (0.165 − 0.508i)13-s + (1.17 − 0.851i)14-s + (0.180 + 0.130i)15-s + (−0.201 − 0.619i)16-s + (−0.255 − 0.787i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.73742 - 1.04534i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.73742 - 1.04534i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-21.8 + 15.8i)T \) |
| 11 | \( 1 + (3.08e3 - 3.15e3i)T \) |
good | 2 | \( 1 + (-2.98 + 9.17i)T + (-103. - 75.2i)T^{2} \) |
| 5 | \( 1 + (-33.3 - 102. i)T + (-6.32e4 + 4.59e4i)T^{2} \) |
| 7 | \( 1 + (-1.24e3 - 906. i)T + (2.54e5 + 7.83e5i)T^{2} \) |
| 13 | \( 1 + (-1.30e3 + 4.02e3i)T + (-5.07e7 - 3.68e7i)T^{2} \) |
| 17 | \( 1 + (5.18e3 + 1.59e4i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-2.02e4 + 1.47e4i)T + (2.76e8 - 8.50e8i)T^{2} \) |
| 23 | \( 1 + 6.79e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (3.53e3 + 2.56e3i)T + (5.33e9 + 1.64e10i)T^{2} \) |
| 31 | \( 1 + (8.90e4 - 2.74e5i)T + (-2.22e10 - 1.61e10i)T^{2} \) |
| 37 | \( 1 + (2.55e4 + 1.85e4i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (3.96e5 - 2.88e5i)T + (6.01e10 - 1.85e11i)T^{2} \) |
| 43 | \( 1 + 6.11e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-5.62e5 + 4.08e5i)T + (1.56e11 - 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-4.82e5 + 1.48e6i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (8.19e5 + 5.95e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (-6.25e5 - 1.92e6i)T + (-2.54e12 + 1.84e12i)T^{2} \) |
| 67 | \( 1 + 8.08e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (6.03e5 + 1.85e6i)T + (-7.35e12 + 5.34e12i)T^{2} \) |
| 73 | \( 1 + (4.63e6 + 3.36e6i)T + (3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (1.65e6 - 5.08e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (2.17e5 + 6.69e5i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + 9.27e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-3.67e6 + 1.13e7i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−14.88201287025991793701012015476, −13.67308459570569539260109808689, −12.36603796991946941567711881076, −11.52403998726993009121764781328, −10.24328504335331733675646407055, −8.430611617722585099109923694931, −7.20642358916072239218810022796, −4.98156672797009249402361635898, −2.86365398124597818852242350324, −1.78698156716873867342609728833,
1.63260436675609765284483608514, 4.27303302394938681982819157674, 5.61493763220439223349504597049, 7.48793071369316911871596396680, 8.384938715315593218346049368605, 10.36819759957167297243821204667, 11.34119194646770136985377094910, 13.51766374739091570689481473831, 14.21307523026651354040649874097, 15.22804045291054043364486332511