| L(s) = 1 | + (9.93 − 6.38i)2-s + (−2.21 + 15.4i)3-s + (31.3 − 68.6i)4-s + (41.1 + 140. i)5-s + (76.4 + 167. i)6-s + (−16.6 + 14.4i)7-s + (−19.3 − 134. i)8-s + (−233. − 68.4i)9-s + (1.30e3 + 1.13e3i)10-s + (−711. + 1.10e3i)11-s + (990. + 636. i)12-s + (−1.27e3 + 1.46e3i)13-s + (−73.2 + 249. i)14-s + (−2.25e3 + 324. i)15-s + (2.11e3 + 2.43e3i)16-s + (3.88e3 − 1.77e3i)17-s + ⋯ |
| L(s) = 1 | + (1.24 − 0.798i)2-s + (−0.0821 + 0.571i)3-s + (0.490 − 1.07i)4-s + (0.329 + 1.12i)5-s + (0.354 + 0.775i)6-s + (−0.0484 + 0.0420i)7-s + (−0.0378 − 0.263i)8-s + (−0.319 − 0.0939i)9-s + (1.30 + 1.13i)10-s + (−0.534 + 0.831i)11-s + (0.573 + 0.368i)12-s + (−0.578 + 0.667i)13-s + (−0.0266 + 0.0908i)14-s + (−0.668 + 0.0961i)15-s + (0.515 + 0.595i)16-s + (0.791 − 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.07442 + 1.33680i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.07442 + 1.33680i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.21 - 15.4i)T \) |
| 23 | \( 1 + (6.98e3 + 9.95e3i)T \) |
| good | 2 | \( 1 + (-9.93 + 6.38i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (-41.1 - 140. i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (16.6 - 14.4i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (711. - 1.10e3i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (1.27e3 - 1.46e3i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (-3.88e3 + 1.77e3i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (-1.08e4 - 4.93e3i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (3.72e3 + 8.16e3i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (2.52e3 + 1.75e4i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (-8.67e3 + 2.95e4i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (3.92e4 - 1.15e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (5.86e4 + 8.43e3i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 - 1.21e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (3.74e4 - 3.24e4i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (8.39e4 - 9.68e4i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (-3.05e5 + 4.39e4i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (1.73e5 + 2.70e5i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (6.76e4 - 4.34e4i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (-2.63e5 + 5.77e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (-2.26e4 - 1.96e4i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (-2.02e5 + 6.90e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (9.77e5 + 1.40e5i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (-3.08e5 - 1.05e6i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−13.88337747046573564479260026211, −12.40516987464812926877134416944, −11.61479431233898018064995851753, −10.44101349142943275360995520556, −9.740009183030696948901665935528, −7.50605812069850609059010699966, −5.91931813528508029874985768328, −4.71220559786283795752102376278, −3.33729854748374983651576127824, −2.23570662597879480046208306687,
0.931275424267702635321829871553, 3.26675605712355367428080208852, 5.16871922922025960762893270184, 5.60341086339525751665801281306, 7.19245100098659194268539430464, 8.283619104983975785885277795772, 9.859014652171529427810711527513, 11.75478711048536816019194949762, 12.71840839043073417063874929488, 13.40541405097514482116785425938
