| L(s) = 1 | + (0.274 − 2.81i)2-s + (2.56 + 4.51i)3-s + (−7.84 − 1.54i)4-s + (2.35 + 6.47i)5-s + (13.4 − 5.97i)6-s + (6.19 − 1.09i)7-s + (−6.51 + 21.6i)8-s + (−13.8 + 23.1i)9-s + (18.8 − 4.85i)10-s + (20.4 + 7.42i)11-s + (−13.1 − 39.4i)12-s + (59.5 + 49.9i)13-s + (−1.37 − 17.7i)14-s + (−23.2 + 27.2i)15-s + (59.2 + 24.2i)16-s + (34.1 + 19.6i)17-s + ⋯ |
| L(s) = 1 | + (0.0971 − 0.995i)2-s + (0.493 + 0.869i)3-s + (−0.981 − 0.193i)4-s + (0.210 + 0.578i)5-s + (0.913 − 0.406i)6-s + (0.334 − 0.0590i)7-s + (−0.287 + 0.957i)8-s + (−0.512 + 0.858i)9-s + (0.596 − 0.153i)10-s + (0.559 + 0.203i)11-s + (−0.316 − 0.948i)12-s + (1.27 + 1.06i)13-s + (−0.0262 − 0.338i)14-s + (−0.399 + 0.469i)15-s + (0.925 + 0.379i)16-s + (0.486 + 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.357i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.934 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.80677 + 0.333761i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80677 + 0.333761i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.274 + 2.81i)T \) |
| 3 | \( 1 + (-2.56 - 4.51i)T \) |
| good | 5 | \( 1 + (-2.35 - 6.47i)T + (-95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-6.19 + 1.09i)T + (322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-20.4 - 7.42i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (-59.5 - 49.9i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-34.1 - 19.6i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (102. - 59.4i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-29.8 + 169. i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-13.5 - 16.1i)T + (-4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (123. + 21.7i)T + (2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (100. - 173. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-154. + 184. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-20.3 + 55.7i)T + (-6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (45.4 + 257. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + 26.8iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (104. - 37.8i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-105. - 597. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-506. + 603. i)T + (-5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-76.4 + 132. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (273. + 472. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (103. + 123. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-889. + 746. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-1.20e3 + 697. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.25e3 - 457. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−13.37412222191078708792242007587, −12.05805755114138842621323799690, −10.84621765900818596859189476738, −10.41983685721668832621486391799, −9.096012619431608316780859328174, −8.376391964161487618131143427601, −6.27046233404411122772397162896, −4.54718742906858082809743225966, −3.58840078057000900116901901599, −1.99168509874684646306940967114,
1.07411651818705585789544938077, 3.55365300448856235066850070594, 5.36025394088591517438559218161, 6.41974183365601617422736822757, 7.68228139231591459140635184401, 8.580845081614462458116225102853, 9.337953771391524316166685539236, 11.21343492851745021262194097087, 12.74065461015147352962074941996, 13.17341489824300447510794198188
