L(s) = 1 | + (−2.26 + 1.64i)2-s + (−0.309 − 0.951i)3-s + (1.80 − 5.54i)4-s + (2.68 + 1.95i)5-s + (2.26 + 1.64i)6-s + (−0.449 + 1.38i)7-s + (3.31 + 10.1i)8-s + (−0.809 + 0.587i)9-s − 9.28·10-s + (0.682 − 3.24i)11-s − 5.83·12-s + (0.809 − 0.587i)13-s + (−1.25 − 3.86i)14-s + (1.02 − 3.15i)15-s + (−14.8 − 10.7i)16-s + (5.09 + 3.70i)17-s + ⋯ |
L(s) = 1 | + (−1.60 + 1.16i)2-s + (−0.178 − 0.549i)3-s + (0.901 − 2.77i)4-s + (1.20 + 0.872i)5-s + (0.924 + 0.671i)6-s + (−0.169 + 0.522i)7-s + (1.17 + 3.60i)8-s + (−0.269 + 0.195i)9-s − 2.93·10-s + (0.205 − 0.978i)11-s − 1.68·12-s + (0.224 − 0.163i)13-s + (−0.335 − 1.03i)14-s + (0.264 − 0.815i)15-s + (−3.71 − 2.69i)16-s + (1.23 + 0.898i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.592313 + 0.469177i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.592313 + 0.469177i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 + (-0.682 + 3.24i)T \) |
| 13 | \( 1 + (-0.809 + 0.587i)T \) |
good | 2 | \( 1 + (2.26 - 1.64i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (-2.68 - 1.95i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (0.449 - 1.38i)T + (-5.66 - 4.11i)T^{2} \) |
| 17 | \( 1 + (-5.09 - 3.70i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.174 - 0.537i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 3.52T + 23T^{2} \) |
| 29 | \( 1 + (0.493 - 1.51i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-7.96 + 5.78i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.422 - 1.30i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.12 - 3.47i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 4.12T + 43T^{2} \) |
| 47 | \( 1 + (-1.75 - 5.41i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.89 + 1.37i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (2.82 - 8.68i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.21 - 1.60i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 6.27T + 67T^{2} \) |
| 71 | \( 1 + (0.143 + 0.104i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.01 - 3.11i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.0831 + 0.0603i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-11.2 - 8.19i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 1.06T + 89T^{2} \) |
| 97 | \( 1 + (-11.8 + 8.57i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.86349060905718538421963950323, −10.17504512775640879111993946707, −9.512749605008167418791407668491, −8.458226320623120945178705196328, −7.80190797424140161147225893287, −6.61436365141599374968178595361, −6.00114942313065243293797232256, −5.63671439543295186353824659678, −2.59153326446434694038300931785, −1.29843515128207536141016673227,
0.996872940391275884506708284584, 2.21690021937030744643466605786, 3.65020428882172433960295392113, 4.92495746773500330694673619732, 6.56627563632243746672126856632, 7.66546343903675210909890276588, 8.728840565744443136945654529288, 9.482723985119717842913166604697, 10.00163085926394836541197584927, 10.46481482652928011975663529281