L(s) = 1 | + (−1 − i)2-s + (−2.28 − 1.94i)3-s + 2i·4-s + (4.58 − 1.99i)5-s + (0.347 + 4.22i)6-s + (5.57 + 4.23i)7-s + (2 − 2i)8-s + (1.46 + 8.87i)9-s + (−6.58 − 2.58i)10-s + 14.6i·11-s + (3.88 − 4.57i)12-s + (−3.48 + 3.48i)13-s + (−1.34 − 9.80i)14-s + (−14.3 − 4.32i)15-s − 4·16-s + (20.1 − 20.1i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (−0.762 − 0.646i)3-s + 0.5i·4-s + (0.916 − 0.399i)5-s + (0.0579 + 0.704i)6-s + (0.796 + 0.604i)7-s + (0.250 − 0.250i)8-s + (0.163 + 0.986i)9-s + (−0.658 − 0.258i)10-s + 1.32i·11-s + (0.323 − 0.381i)12-s + (−0.267 + 0.267i)13-s + (−0.0961 − 0.700i)14-s + (−0.957 − 0.288i)15-s − 0.250·16-s + (1.18 − 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.12788 - 0.418125i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12788 - 0.418125i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (2.28 + 1.94i)T \) |
| 5 | \( 1 + (-4.58 + 1.99i)T \) |
| 7 | \( 1 + (-5.57 - 4.23i)T \) |
good | 11 | \( 1 - 14.6iT - 121T^{2} \) |
| 13 | \( 1 + (3.48 - 3.48i)T - 169iT^{2} \) |
| 17 | \( 1 + (-20.1 + 20.1i)T - 289iT^{2} \) |
| 19 | \( 1 - 26.4T + 361T^{2} \) |
| 23 | \( 1 + (2.68 - 2.68i)T - 529iT^{2} \) |
| 29 | \( 1 + 28.5T + 841T^{2} \) |
| 31 | \( 1 + 15.6iT - 961T^{2} \) |
| 37 | \( 1 + (-7.69 + 7.69i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 37.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-41.7 - 41.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (21.0 - 21.0i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-47.4 + 47.4i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 61.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 54.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-68.9 + 68.9i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 65.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-6.51 + 6.51i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 42.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (9.52 + 9.52i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 19.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (84.6 + 84.6i)T + 9.40e3iT^{2} \) |
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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
−11.99500432641393137413933833809, −11.28691118303670832544563402078, −9.907498099915475979544312666862, −9.419797412214707978346051649150, −7.909749294959012797161173845968, −7.12942985548933972231771626230, −5.58022730241305875763729016864, −4.85817532045905542630718240975, −2.36887319920942557014381728669, −1.29344790169983435542084387050,
1.14195176532973354055453107320, 3.55733235894958434362751342784, 5.33032932984853710369213303163, 5.80819690098104184367731829677, 7.08826534212976577331152816283, 8.279696199884010117000220216673, 9.482387043907186201661922029410, 10.37866537129761438602242419735, 10.88904649601049128755281724993, 11.94679955919377379254740708486