L(s) = 1 | − 0.0683·2-s − 1.99·4-s + (−1.33 + 2.30i)5-s + 0.273·8-s + (0.0910 − 0.157i)10-s + (−0.799 − 1.38i)11-s + (−2.62 − 4.54i)13-s + 3.97·16-s + (−3.27 + 5.67i)17-s + (−0.950 − 1.64i)19-s + (2.65 − 4.60i)20-s + (0.0546 + 0.0946i)22-s + (−1.53 + 2.65i)23-s + (−1.04 − 1.81i)25-s + (0.179 + 0.311i)26-s + ⋯ |
L(s) = 1 | − 0.0483·2-s − 0.997·4-s + (−0.595 + 1.03i)5-s + 0.0965·8-s + (0.0287 − 0.0498i)10-s + (−0.241 − 0.417i)11-s + (−0.728 − 1.26i)13-s + 0.992·16-s + (−0.793 + 1.37i)17-s + (−0.218 − 0.377i)19-s + (0.594 − 1.02i)20-s + (0.0116 + 0.0201i)22-s + (−0.319 + 0.554i)23-s + (−0.209 − 0.363i)25-s + (0.0352 + 0.0610i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7324505234\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7324505234\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.0683T + 2T^{2} \) |
| 5 | \( 1 + (1.33 - 2.30i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.799 + 1.38i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.62 + 4.54i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.27 - 5.67i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.950 + 1.64i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.53 - 2.65i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.19 + 5.53i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.71T + 31T^{2} \) |
| 37 | \( 1 + (2.11 + 3.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.69 - 6.40i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.63 + 9.75i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.79T + 47T^{2} \) |
| 53 | \( 1 + (-4.44 + 7.70i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 2.71T + 61T^{2} \) |
| 67 | \( 1 + 3.32T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + (1.09 - 1.90i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 0.813T + 79T^{2} \) |
| 83 | \( 1 + (-3.41 + 5.92i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.235 - 0.407i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.57 + 4.46i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.694780903239857575800306499128, −8.467870485585430244251090697885, −8.112509384914403640628961642312, −7.21648376713623885415720948369, −6.20073366461609505043550011442, −5.33526709190742730398106367372, −4.26073763680689215689410334553, −3.49903261696395157599044911792, −2.48340879782760392355587889417, −0.43907177331081509611807161188,
0.895048046220869901128026116865, 2.49588179629192295646292481938, 4.03375183258825492241180581027, 4.67814827663286711385606472209, 5.06163491205261803325582415370, 6.50088787986993214123899650447, 7.42334555946882976571360558681, 8.302515009569689314399500802244, 8.929983161046008480985010789931, 9.500552329888280279025321816093