L(s) = 1 | + 5.11·2-s + (−1.51 + 4.96i)3-s + 18.1·4-s + (10.9 − 2.25i)5-s + (−7.75 + 25.4i)6-s + (−11.4 − 14.5i)7-s + 51.7·8-s + (−22.3 − 15.0i)9-s + (55.9 − 11.5i)10-s + 55.4i·11-s + (−27.5 + 90.0i)12-s − 20.9·13-s + (−58.6 − 74.3i)14-s + (−5.42 + 57.8i)15-s + 119.·16-s − 96.8i·17-s + ⋯ |
L(s) = 1 | + 1.80·2-s + (−0.292 + 0.956i)3-s + 2.26·4-s + (0.979 − 0.201i)5-s + (−0.527 + 1.72i)6-s + (−0.619 − 0.785i)7-s + 2.28·8-s + (−0.829 − 0.558i)9-s + (1.77 − 0.364i)10-s + 1.51i·11-s + (−0.661 + 2.16i)12-s − 0.447·13-s + (−1.11 − 1.41i)14-s + (−0.0933 + 0.995i)15-s + 1.86·16-s − 1.38i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.92096 + 1.15799i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.92096 + 1.15799i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.51 - 4.96i)T \) |
| 5 | \( 1 + (-10.9 + 2.25i)T \) |
| 7 | \( 1 + (11.4 + 14.5i)T \) |
good | 2 | \( 1 - 5.11T + 8T^{2} \) |
| 11 | \( 1 - 55.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 20.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 96.8iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 33.1iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 156. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 80.4iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 180. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 36.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 52.5iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 259. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 191.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 705.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 427. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 306. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 513. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 360.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 85.9T + 4.93e5T^{2} \) |
| 83 | \( 1 - 886. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−13.53259271673583953458174848119, −12.49574730185969622649848906084, −11.62337963435198229724700148148, −10.17234380210433699046536792454, −9.700877086993782569284439776117, −7.17124992618384320746669459779, −6.08104374150967472487688056558, −4.95903143769330045744242273114, −4.12399331651657799565421463544, −2.54843974325741529001088802838,
2.08527548704675770204588262163, 3.26534195475109435053125590816, 5.45673676846181837675234291362, 5.98778002068121960964736712750, 6.82132053070338809061312802372, 8.596496217884043999306718872678, 10.52727676253206112118378398337, 11.56009223321651725216245731585, 12.58817649403329515538784082238, 13.12290927160941818760955967137