L(s) = 1 | + (1.36 + 0.366i)2-s + (−1.67 + 0.448i)3-s + (1.73 + i)4-s + (3.95 + 3.06i)5-s − 2.44·6-s + (3.71 − 5.93i)7-s + (1.99 + 2i)8-s + (2.59 − 1.50i)9-s + (4.27 + 5.63i)10-s + (−3.58 + 6.20i)11-s + (−3.34 − 0.896i)12-s + (7.28 + 7.28i)13-s + (7.24 − 6.74i)14-s + (−7.98 − 3.35i)15-s + (1.99 + 3.46i)16-s + (4.87 + 18.1i)17-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (−0.557 + 0.149i)3-s + (0.433 + 0.250i)4-s + (0.790 + 0.612i)5-s − 0.408·6-s + (0.530 − 0.847i)7-s + (0.249 + 0.250i)8-s + (0.288 − 0.166i)9-s + (0.427 + 0.563i)10-s + (−0.325 + 0.564i)11-s + (−0.278 − 0.0747i)12-s + (0.560 + 0.560i)13-s + (0.517 − 0.481i)14-s + (−0.532 − 0.223i)15-s + (0.124 + 0.216i)16-s + (0.286 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.11116 + 0.870383i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.11116 + 0.870383i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 3 | \( 1 + (1.67 - 0.448i)T \) |
| 5 | \( 1 + (-3.95 - 3.06i)T \) |
| 7 | \( 1 + (-3.71 + 5.93i)T \) |
good | 11 | \( 1 + (3.58 - 6.20i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-7.28 - 7.28i)T + 169iT^{2} \) |
| 17 | \( 1 + (-4.87 - 18.1i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-7.65 + 4.41i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (0.107 - 0.401i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 + 27.7iT - 841T^{2} \) |
| 31 | \( 1 + (-12.5 + 21.6i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (51.5 + 13.8i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 - 46.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + (37.3 + 37.3i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (31.0 + 8.32i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (68.8 - 18.4i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (35.2 + 20.3i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-3.30 - 5.72i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (34.4 + 128. i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 129.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (7.85 - 2.10i)T + (4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (59.9 - 34.6i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-57.0 - 57.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-114. + 65.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (46.0 - 46.0i)T - 9.40e3iT^{2} \) |
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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
−12.32167442292359033320894430219, −11.18369538428348005854828825053, −10.57669562347444431260544690139, −9.616163556761254583027265782412, −7.947524835510894075164929498042, −6.87905928483807615905341316574, −6.01978787052990814458638320525, −4.86953614055695266506452618640, −3.69506788042334696534113670124, −1.82803506290021865480101859633,
1.35965905608254561319358197464, 2.97432484503697043592341099469, 4.96396750425522467555258516155, 5.45428756071090448911067490107, 6.46520357781007081445046046039, 8.015253508557655377656641815015, 9.110852459489747763380106889810, 10.28365685178596218741887794862, 11.27145034283595537522943134785, 12.12854475848352719849192469393