| L(s) = 1 | + (7.98 + 4.15i)3-s − 11.1i·5-s − 61.6·7-s + (46.4 + 66.3i)9-s − 108. i·11-s − 63.7·13-s + (46.4 − 89.2i)15-s − 175. i·17-s − 301.·19-s + (−491. − 255. i)21-s − 1.03e3i·23-s − 125.·25-s + (95.6 + 722. i)27-s − 179. i·29-s − 1.06e3·31-s + ⋯ |
| L(s) = 1 | + (0.887 + 0.461i)3-s − 0.447i·5-s − 1.25·7-s + (0.573 + 0.818i)9-s − 0.899i·11-s − 0.376·13-s + (0.206 − 0.396i)15-s − 0.606i·17-s − 0.835·19-s + (−1.11 − 0.580i)21-s − 1.96i·23-s − 0.200·25-s + (0.131 + 0.991i)27-s − 0.213i·29-s − 1.11·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.461 + 0.887i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.094101729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.094101729\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-7.98 - 4.15i)T \) |
| 5 | \( 1 + 11.1iT \) |
| good | 7 | \( 1 + 61.6T + 2.40e3T^{2} \) |
| 11 | \( 1 + 108. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 63.7T + 2.85e4T^{2} \) |
| 17 | \( 1 + 175. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 301.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 1.03e3iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 179. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.06e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.06e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 173. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.03e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 935. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 423. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.18e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.30e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 4.21e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 475. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.14e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.10e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 3.76e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 8.26e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.31e3T + 8.85e7T^{2} \) |
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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
−10.90184853304727308939836698528, −10.06345230428797109884794791116, −9.100967692810105952699489957400, −8.522241698732452293223107569671, −7.23043331726895555774631009711, −6.05389638572765934749500325882, −4.63505059987487544938547710498, −3.48886977480571015415626426899, −2.41628775205237883645708287980, −0.30037588920490447875992941355,
1.79545989661179870178733797400, 3.04270325478890121287274620262, 4.03583753127688042012503866248, 5.92039957444893918404244361141, 6.99367960350332673774016730001, 7.61685217665969893505359980543, 9.044681821971163166816954743546, 9.669935223620479410834134985767, 10.61903993592828030537191230109, 12.10238337353968986527412286251
