L(s) = 1 | + 5.51i·2-s + 3.00·3-s − 22.4·4-s − 18.8i·5-s + 16.5i·6-s − 0.324i·7-s − 79.5i·8-s − 17.9·9-s + 104.·10-s + 11i·11-s − 67.3·12-s + (2.96 − 46.7i)13-s + 1.79·14-s − 56.7i·15-s + 259.·16-s + 85.7·17-s + ⋯ |
L(s) = 1 | + 1.94i·2-s + 0.578·3-s − 2.80·4-s − 1.68i·5-s + 1.12i·6-s − 0.0175i·7-s − 3.51i·8-s − 0.665·9-s + 3.29·10-s + 0.301i·11-s − 1.62·12-s + (0.0632 − 0.997i)13-s + 0.0342·14-s − 0.976i·15-s + 4.05·16-s + 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0632i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.08563 - 0.0343483i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08563 - 0.0343483i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - 11iT \) |
| 13 | \( 1 + (-2.96 + 46.7i)T \) |
good | 2 | \( 1 - 5.51iT - 8T^{2} \) |
| 3 | \( 1 - 3.00T + 27T^{2} \) |
| 5 | \( 1 + 18.8iT - 125T^{2} \) |
| 7 | \( 1 + 0.324iT - 343T^{2} \) |
| 17 | \( 1 - 85.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 62.7iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 265. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 120. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 49.7iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 38.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 94.6iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 237.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 282. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 359.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 325. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 658. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 781.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 822. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 976. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.76e3iT - 9.12e5T^{2} \) |
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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
−13.14992822969097563166513149933, −12.18731819414942531054397941299, −9.787964964530539320327284609095, −9.053738909939842620832725195009, −8.168868568094892773385941695299, −7.66762427864037510660113499757, −5.84312119205885434071294117257, −5.22845456593470033429614796821, −3.94620617557181293254103198123, −0.51028911261946340220709631819,
1.97461385250650625841094943174, 3.09577601753574370930332027338, 3.79914191494811583142713307842, 5.79766335878283437911478681969, 7.71263567176796801940011629536, 8.912885999401738223924181954915, 9.942806521757023543990266186433, 10.69777884166504190405402688975, 11.52948735223525285577028892441, 12.27520362246575764872547646871