L(s) = 1 | + (−1.44 + 1.04i)2-s + (−0.309 − 0.951i)3-s + (0.362 − 1.11i)4-s + (2.24 + 1.63i)5-s + (1.44 + 1.04i)6-s + (−0.0806 + 0.248i)7-s + (−0.454 − 1.39i)8-s + (−0.809 + 0.587i)9-s − 4.95·10-s + (0.650 + 3.25i)11-s − 1.17·12-s + (−0.809 + 0.587i)13-s + (−0.143 − 0.442i)14-s + (0.859 − 2.64i)15-s + (4.02 + 2.92i)16-s + (3.03 + 2.20i)17-s + ⋯ |
L(s) = 1 | + (−1.01 + 0.740i)2-s + (−0.178 − 0.549i)3-s + (0.181 − 0.558i)4-s + (1.00 + 0.731i)5-s + (0.588 + 0.427i)6-s + (−0.0304 + 0.0938i)7-s + (−0.160 − 0.494i)8-s + (−0.269 + 0.195i)9-s − 1.56·10-s + (0.196 + 0.980i)11-s − 0.339·12-s + (−0.224 + 0.163i)13-s + (−0.0384 − 0.118i)14-s + (0.221 − 0.682i)15-s + (1.00 + 0.730i)16-s + (0.736 + 0.534i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.534248 + 0.636110i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.534248 + 0.636110i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.650 - 3.25i)T \) |
| 13 | \( 1 + (0.809 - 0.587i)T \) |
good | 2 | \( 1 + (1.44 - 1.04i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (-2.24 - 1.63i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (0.0806 - 0.248i)T + (-5.66 - 4.11i)T^{2} \) |
| 17 | \( 1 + (-3.03 - 2.20i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.80 + 5.56i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 7.50T + 23T^{2} \) |
| 29 | \( 1 + (2.17 - 6.69i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (7.33 - 5.32i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.58 - 7.95i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.24 - 3.81i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 1.53T + 43T^{2} \) |
| 47 | \( 1 + (-2.73 - 8.40i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.20 + 0.875i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.10 + 6.49i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (8.26 + 6.00i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 8.40T + 67T^{2} \) |
| 71 | \( 1 + (5.01 + 3.64i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.191 + 0.589i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-6.87 + 4.99i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.59 + 1.16i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + (-9.86 + 7.16i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.06852633319640967057447753733, −10.32272337179293563870986641748, −9.394479941729095349041993847940, −8.804044478878954122185468544104, −7.46192930974284438732562770640, −6.93715297819652880545325702979, −6.24551939885123933858300056415, −5.01863438517081674514019062094, −3.05189348141773058204820784909, −1.55824883534330690055867086184,
0.814556874584643315195165288122, 2.23805609432712055173317192982, 3.71708612220188466229653174015, 5.44087279579025795418397743350, 5.73669995026984098163183203496, 7.55200027250089991590566865483, 8.748976362915326623815845150023, 9.199344158621789585901164921117, 9.980772541286158109651020768710, 10.67108734278207395901662661188