| L(s) = 1 | + (0.467 − 1.33i)2-s + (3.04 + 0.342i)3-s + (−1.56 − 1.24i)4-s + (−2.27 − 4.72i)5-s + (1.87 − 3.90i)6-s + (6.13 + 7.69i)7-s + (−2.39 + 1.50i)8-s + (0.365 + 0.0834i)9-s + (−7.36 + 0.829i)10-s + (−0.606 − 0.381i)11-s + (−4.33 − 4.33i)12-s + (−3.76 + 0.858i)13-s + (13.1 − 4.59i)14-s + (−5.29 − 15.1i)15-s + (0.890 + 3.89i)16-s + (−7.17 + 7.17i)17-s + ⋯ |
| L(s) = 1 | + (0.233 − 0.667i)2-s + (1.01 + 0.114i)3-s + (−0.390 − 0.311i)4-s + (−0.454 − 0.944i)5-s + (0.313 − 0.650i)6-s + (0.876 + 1.09i)7-s + (−0.299 + 0.188i)8-s + (0.0406 + 0.00927i)9-s + (−0.736 + 0.0829i)10-s + (−0.0551 − 0.0346i)11-s + (−0.360 − 0.360i)12-s + (−0.289 + 0.0660i)13-s + (0.938 − 0.328i)14-s + (−0.353 − 1.00i)15-s + (0.0556 + 0.243i)16-s + (−0.421 + 0.421i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.41603 - 0.694621i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41603 - 0.694621i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.467 + 1.33i)T \) |
| 29 | \( 1 + (-26.6 - 11.3i)T \) |
| good | 3 | \( 1 + (-3.04 - 0.342i)T + (8.77 + 2.00i)T^{2} \) |
| 5 | \( 1 + (2.27 + 4.72i)T + (-15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (-6.13 - 7.69i)T + (-10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (0.606 + 0.381i)T + (52.4 + 109. i)T^{2} \) |
| 13 | \( 1 + (3.76 - 0.858i)T + (152. - 73.3i)T^{2} \) |
| 17 | \( 1 + (7.17 - 7.17i)T - 289iT^{2} \) |
| 19 | \( 1 + (-3.50 - 31.1i)T + (-351. + 80.3i)T^{2} \) |
| 23 | \( 1 + (16.2 + 7.81i)T + (329. + 413. i)T^{2} \) |
| 31 | \( 1 + (-19.8 + 56.7i)T + (-751. - 599. i)T^{2} \) |
| 37 | \( 1 + (22.3 - 14.0i)T + (593. - 1.23e3i)T^{2} \) |
| 41 | \( 1 + (39.4 + 39.4i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (12.9 - 4.52i)T + (1.44e3 - 1.15e3i)T^{2} \) |
| 47 | \( 1 + (-38.1 + 60.6i)T + (-958. - 1.99e3i)T^{2} \) |
| 53 | \( 1 + (-57.7 + 27.7i)T + (1.75e3 - 2.19e3i)T^{2} \) |
| 59 | \( 1 - 76.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + (47.5 + 5.36i)T + (3.62e3 + 828. i)T^{2} \) |
| 67 | \( 1 + (48.6 + 11.1i)T + (4.04e3 + 1.94e3i)T^{2} \) |
| 71 | \( 1 + (31.9 - 7.30i)T + (4.54e3 - 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-46.6 - 133. i)T + (-4.16e3 + 3.32e3i)T^{2} \) |
| 79 | \( 1 + (-10.1 - 16.1i)T + (-2.70e3 + 5.62e3i)T^{2} \) |
| 83 | \( 1 + (72.1 - 90.5i)T + (-1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (25.4 - 72.8i)T + (-6.19e3 - 4.93e3i)T^{2} \) |
| 97 | \( 1 + (-54.0 + 6.08i)T + (9.17e3 - 2.09e3i)T^{2} \) |
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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
−14.68986663977863675491108098800, −13.69758294048727108036388365537, −12.30637648340653745684306049701, −11.74870854683560745621632131286, −9.997272520473425897710336033581, −8.561526044972159609754853033378, −8.301750925411933678536186613266, −5.51200693789875536999929703771, −4.04409052149761158967042065891, −2.17728442643556032407770659713,
3.04545536426954450976636335114, 4.64598630948240899581293178001, 6.93314170381279624353816288393, 7.66604709166040852966268437231, 8.810078643507037553609967605580, 10.48516130763967202911233279781, 11.64911134599270905072031582978, 13.55102441551466667627329636858, 14.04562758134411942541161662791, 14.91407860010270232543997688011
