| L(s) = 1 | + (1.72 + 1.00i)2-s + (1.35 + 1.07i)3-s + (1.96 + 3.48i)4-s + (−8.48 − 4.08i)5-s + (1.25 + 3.23i)6-s + (−5.64 − 4.50i)7-s + (−0.115 + 7.99i)8-s + (0.667 + 2.92i)9-s + (−10.5 − 15.6i)10-s + (−17.1 − 3.91i)11-s + (−1.09 + 6.84i)12-s + (−2.74 + 12.0i)13-s + (−5.21 − 13.4i)14-s + (−7.07 − 14.6i)15-s + (−8.26 + 13.7i)16-s + 11.3·17-s + ⋯ |
| L(s) = 1 | + (0.863 + 0.504i)2-s + (0.451 + 0.359i)3-s + (0.491 + 0.870i)4-s + (−1.69 − 0.816i)5-s + (0.208 + 0.538i)6-s + (−0.806 − 0.643i)7-s + (−0.0144 + 0.999i)8-s + (0.0741 + 0.324i)9-s + (−1.05 − 1.56i)10-s + (−1.56 − 0.356i)11-s + (−0.0915 + 0.570i)12-s + (−0.211 + 0.924i)13-s + (−0.372 − 0.961i)14-s + (−0.471 − 0.979i)15-s + (−0.516 + 0.856i)16-s + 0.665·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.482i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0785892 - 0.305532i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0785892 - 0.305532i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.72 - 1.00i)T \) |
| 3 | \( 1 + (-1.35 - 1.07i)T \) |
| 29 | \( 1 + (3.42 + 28.7i)T \) |
| good | 5 | \( 1 + (8.48 + 4.08i)T + (15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (5.64 + 4.50i)T + (10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (17.1 + 3.91i)T + (109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (2.74 - 12.0i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 - 11.3T + 289T^{2} \) |
| 19 | \( 1 + (21.6 - 17.2i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (3.18 + 6.61i)T + (-329. + 413. i)T^{2} \) |
| 31 | \( 1 + (-18.3 + 38.1i)T + (-599. - 751. i)T^{2} \) |
| 37 | \( 1 + (-7.17 - 31.4i)T + (-1.23e3 + 593. i)T^{2} \) |
| 41 | \( 1 - 16.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-4.08 - 8.47i)T + (-1.15e3 + 1.44e3i)T^{2} \) |
| 47 | \( 1 + (-76.8 - 17.5i)T + (1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (31.3 + 15.0i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 + 12.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + (6.57 - 8.23i)T + (-828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (27.3 - 6.24i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (107. + 24.5i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (111. - 53.5i)T + (3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 + (79.6 - 18.1i)T + (5.62e3 - 2.70e3i)T^{2} \) |
| 83 | \( 1 + (128. - 102. i)T + (1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (60.4 + 29.1i)T + (4.93e3 + 6.19e3i)T^{2} \) |
| 97 | \( 1 + (-1.63 - 2.05i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−12.02726910470864314862700018739, −11.12592509150835104618803689230, −10.03278457616821578595077490886, −8.568433089803550682406345574464, −7.947541572976667491537348825801, −7.28636099702964061294995362817, −5.86076017095767053446074553817, −4.44153530595074988508928780325, −4.05513858686745986235823174541, −2.87102355309158230500550562903,
0.095658825782460366380618377768, 2.76395031123521326626633475814, 3.08075018777866147847760593752, 4.42858910636922786928099194499, 5.72820564748550957431953055672, 7.02879059245673835578591640135, 7.61710735224703929041559213630, 8.781911450255257378139669953896, 10.34163738206397524581302245978, 10.73331921042719024604943205665
