| L(s) = 1 | + (−0.633 − 0.134i)2-s + (0.913 − 0.406i)3-s + (−1.44 − 0.642i)4-s + (−0.543 + 1.67i)5-s + (−0.633 + 0.134i)6-s + (−0.789 − 0.351i)7-s + (1.87 + 1.36i)8-s + (0.669 − 0.743i)9-s + (0.569 − 0.986i)10-s + (2.43 + 2.25i)11-s − 1.58·12-s + (1.51 − 3.26i)13-s + (0.452 + 0.329i)14-s + (0.183 + 1.74i)15-s + (1.10 + 1.23i)16-s + (5.36 − 1.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.448 − 0.0952i)2-s + (0.527 − 0.234i)3-s + (−0.721 − 0.321i)4-s + (−0.242 + 0.747i)5-s + (−0.258 + 0.0549i)6-s + (−0.298 − 0.132i)7-s + (0.663 + 0.481i)8-s + (0.223 − 0.247i)9-s + (0.180 − 0.311i)10-s + (0.734 + 0.678i)11-s − 0.456·12-s + (0.421 − 0.906i)13-s + (0.121 + 0.0879i)14-s + (0.0474 + 0.451i)15-s + (0.277 + 0.308i)16-s + (1.30 − 0.276i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13050 - 0.0162578i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13050 - 0.0162578i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 11 | \( 1 + (-2.43 - 2.25i)T \) |
| 13 | \( 1 + (-1.51 + 3.26i)T \) |
| good | 2 | \( 1 + (0.633 + 0.134i)T + (1.82 + 0.813i)T^{2} \) |
| 5 | \( 1 + (0.543 - 1.67i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (0.789 + 0.351i)T + (4.68 + 5.20i)T^{2} \) |
| 17 | \( 1 + (-5.36 + 1.14i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (0.638 - 6.07i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-4.13 + 7.16i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.171 - 1.63i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-2.00 - 6.15i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.736 + 7.00i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-0.601 + 0.267i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-1.03 - 1.79i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.01 + 0.738i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.74 - 11.5i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-4.74 - 2.11i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (6.74 - 1.43i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (0.135 - 0.234i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.6 + 2.26i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (5.93 - 4.31i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (0.681 + 2.09i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.0983 - 0.302i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.06 + 1.85i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.03 - 7.81i)T + (-10.1 - 96.4i)T^{2} \) |
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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
−10.70517029532708206073234891827, −10.29125961217670137027300180907, −9.363336349476362191161354250843, −8.466739695784845062551015421727, −7.63334327660075703167518741540, −6.68280060412751818915438003386, −5.44214695976922545403061323697, −4.06014110043063316006675163820, −3.03598885830007152960294124112, −1.23694925167689575624155415538,
1.09649016314854294369992220334, 3.27785160192056570781237808656, 4.18444086217734999918995863861, 5.20554417323558671224806449747, 6.66575147115509104322050213643, 7.83955548048592735025391454631, 8.606372290049974686750173063396, 9.253927357759006629107555631669, 9.782022306808652404106761403484, 11.19615002719372324489074234200
