L(s) = 1 | + (0.312 − 1.97i)2-s + (2.75 − 1.18i)3-s + (−3.80 − 1.23i)4-s + (0.00985 + 0.00985i)5-s + (−1.48 − 5.81i)6-s + 6.42i·7-s + (−3.62 + 7.13i)8-s + (6.19 − 6.53i)9-s + (0.0225 − 0.0163i)10-s + (−9.07 − 9.07i)11-s + (−11.9 + 1.10i)12-s + (12.6 + 12.6i)13-s + (12.6 + 2.00i)14-s + (0.0388 + 0.0154i)15-s + (12.9 + 9.39i)16-s + 19.0i·17-s + ⋯ |
L(s) = 1 | + (0.156 − 0.987i)2-s + (0.918 − 0.395i)3-s + (−0.951 − 0.308i)4-s + (0.00197 + 0.00197i)5-s + (−0.246 − 0.969i)6-s + 0.917i·7-s + (−0.453 + 0.891i)8-s + (0.687 − 0.725i)9-s + (0.00225 − 0.00163i)10-s + (−0.824 − 0.824i)11-s + (−0.995 + 0.0923i)12-s + (0.969 + 0.969i)13-s + (0.906 + 0.143i)14-s + (0.00259 + 0.00103i)15-s + (0.809 + 0.586i)16-s + 1.11i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.06052 - 0.892557i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06052 - 0.892557i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.312 + 1.97i)T \) |
| 3 | \( 1 + (-2.75 + 1.18i)T \) |
good | 5 | \( 1 + (-0.00985 - 0.00985i)T + 25iT^{2} \) |
| 7 | \( 1 - 6.42iT - 49T^{2} \) |
| 11 | \( 1 + (9.07 + 9.07i)T + 121iT^{2} \) |
| 13 | \( 1 + (-12.6 - 12.6i)T + 169iT^{2} \) |
| 17 | \( 1 - 19.0iT - 289T^{2} \) |
| 19 | \( 1 + (2.07 + 2.07i)T + 361iT^{2} \) |
| 23 | \( 1 + 19.5T + 529T^{2} \) |
| 29 | \( 1 + (-11.1 + 11.1i)T - 841iT^{2} \) |
| 31 | \( 1 + 59.9T + 961T^{2} \) |
| 37 | \( 1 + (-9.32 + 9.32i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 47.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-24.1 + 24.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 6.29iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (20.6 + 20.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-60.3 - 60.3i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-48.0 - 48.0i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (23.7 + 23.7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 13.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 31.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 47.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-70.3 + 70.3i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 95.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 61.6T + 9.40e3T^{2} \) |
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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
−14.82417970027163931723844974393, −13.79882890634821464817461065480, −12.91756081880385980298571481619, −11.83299274816307046033094557443, −10.47516788106905051595142421448, −8.983797314224881975755273085307, −8.295313908240820392004831680352, −5.96319283527627198099678657733, −3.79371961092650800389969281347, −2.14946129582601885156194617867,
3.60935023540904945344208377328, 5.12568898404102961979249161886, 7.20649707332244917077759988149, 8.063302122908714699208123152679, 9.465484590076420080518627334083, 10.56635651524454623320915603671, 12.89674263829413214146481208244, 13.60018285662308775270155726789, 14.65906920933887274654637409653, 15.65195419808528734949337536806