L(s) = 1 | + (−1.12 − 1.12i)2-s − 3-s + 0.513i·4-s + (0.850 − 0.850i)5-s + (1.12 + 1.12i)6-s + (−0.675 + 0.675i)7-s + (−1.66 + 1.66i)8-s + 9-s − 1.90·10-s + (−0.299 − 3.30i)11-s − 0.513i·12-s + (−3.60 + 0.0438i)13-s + 1.51·14-s + (−0.850 + 0.850i)15-s + 4.76·16-s − 7.43·17-s + ⋯ |
L(s) = 1 | + (−0.792 − 0.792i)2-s − 0.577·3-s + 0.256i·4-s + (0.380 − 0.380i)5-s + (0.457 + 0.457i)6-s + (−0.255 + 0.255i)7-s + (−0.589 + 0.589i)8-s + 0.333·9-s − 0.603·10-s + (−0.0903 − 0.995i)11-s − 0.148i·12-s + (−0.999 + 0.0121i)13-s + 0.404·14-s + (−0.219 + 0.219i)15-s + 1.19·16-s − 1.80·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0338110 + 0.0508161i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0338110 + 0.0508161i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 + (0.299 + 3.30i)T \) |
| 13 | \( 1 + (3.60 - 0.0438i)T \) |
good | 2 | \( 1 + (1.12 + 1.12i)T + 2iT^{2} \) |
| 5 | \( 1 + (-0.850 + 0.850i)T - 5iT^{2} \) |
| 7 | \( 1 + (0.675 - 0.675i)T - 7iT^{2} \) |
| 17 | \( 1 + 7.43T + 17T^{2} \) |
| 19 | \( 1 + (-2.00 - 2.00i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.77iT - 23T^{2} \) |
| 29 | \( 1 + 0.246iT - 29T^{2} \) |
| 31 | \( 1 + (7.23 - 7.23i)T - 31iT^{2} \) |
| 37 | \( 1 + (-0.872 - 0.872i)T + 37iT^{2} \) |
| 41 | \( 1 + (-2.78 - 2.78i)T + 41iT^{2} \) |
| 43 | \( 1 + 4.52T + 43T^{2} \) |
| 47 | \( 1 + (-2.70 - 2.70i)T + 47iT^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + (6.27 + 6.27i)T + 59iT^{2} \) |
| 61 | \( 1 - 3.96iT - 61T^{2} \) |
| 67 | \( 1 + (-6.55 + 6.55i)T - 67iT^{2} \) |
| 71 | \( 1 + (-2.41 + 2.41i)T - 71iT^{2} \) |
| 73 | \( 1 + (-5.88 + 5.88i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.3iT - 79T^{2} \) |
| 83 | \( 1 + (-2.61 - 2.61i)T + 83iT^{2} \) |
| 89 | \( 1 + (5.39 + 5.39i)T + 89iT^{2} \) |
| 97 | \( 1 + (10.7 - 10.7i)T - 97iT^{2} \) |
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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
−10.79792055464769654038431359050, −9.540996234079558451972374752255, −9.139497312180301085636609616596, −8.109745119191153130907444842580, −6.68898895643600042153798974745, −5.72643832939148742959420275942, −4.80209337752547208464507978228, −3.01898067568695670411331230087, −1.70997238423968563283221756085, −0.05103336692113170403188932978,
2.34207543308044732240757886575, 4.10442160876735615179293240876, 5.35337872725553799370452199175, 6.60476790164540142314373320816, 7.03561317530340810302364554419, 7.899384607072026330815204749013, 9.299504647521827532576316712909, 9.668997727879412225528022690691, 10.68299344793547469273696808087, 11.67753855672125149083028352781