| L(s) = 1 | + (5.17 − 0.443i)3-s + (−4.80 + 1.74i)5-s + (5.74 + 32.5i)7-s + (26.6 − 4.59i)9-s + (28.2 + 10.2i)11-s + (7.59 + 6.37i)13-s + (−24.0 + 11.1i)15-s + (44.7 − 77.4i)17-s + (−29.6 − 51.3i)19-s + (44.1 + 166. i)21-s + (−17.0 + 96.7i)23-s + (−75.7 + 63.5i)25-s + (135. − 35.6i)27-s + (108. − 90.6i)29-s + (4.10 − 23.2i)31-s + ⋯ |
| L(s) = 1 | + (0.996 − 0.0854i)3-s + (−0.429 + 0.156i)5-s + (0.310 + 1.75i)7-s + (0.985 − 0.170i)9-s + (0.773 + 0.281i)11-s + (0.162 + 0.136i)13-s + (−0.414 + 0.192i)15-s + (0.638 − 1.10i)17-s + (−0.358 − 0.620i)19-s + (0.459 + 1.72i)21-s + (−0.154 + 0.876i)23-s + (−0.605 + 0.508i)25-s + (0.967 − 0.253i)27-s + (0.692 − 0.580i)29-s + (0.0237 − 0.134i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.03327 + 0.700819i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03327 + 0.700819i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-5.17 + 0.443i)T \) |
| good | 5 | \( 1 + (4.80 - 1.74i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-5.74 - 32.5i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-28.2 - 10.2i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (-7.59 - 6.37i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-44.7 + 77.4i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (29.6 + 51.3i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (17.0 - 96.7i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-108. + 90.6i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-4.10 + 23.2i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (114. - 197. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (357. + 300. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (10.1 + 3.68i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (66.0 + 374. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + 202.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-766. + 279. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (109. + 622. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (466. + 391. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-140. + 243. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-608. - 1.05e3i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-278. + 233. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-453. + 380. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (359. + 622. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-776. - 282. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−13.46502390136515601991849867057, −12.10384195006645296909040931974, −11.63113352904181419284831956066, −9.751870535482748957390758074371, −8.961387633831998327765387726048, −8.044095736058006475200258895934, −6.75312452735898523108827312468, −5.11529334231902112044448522171, −3.42375383075247312142405253573, −2.04044147448313711556515026243,
1.29523173351607659124163616777, 3.60793931378155363800540996685, 4.34022760106375163752974373546, 6.61102180538562702288147841855, 7.80429353682601608985133007402, 8.505865687461009135286439488959, 10.03862029626504400849486482692, 10.71319129228275662924029136088, 12.22851672049519650082153996932, 13.32464045420653718729831522691
