| L(s) = 1 | + (−1.36 + 0.874i)2-s + (−0.318 + 2.21i)3-s + (0.256 − 0.562i)4-s + (−3.10 + 0.911i)5-s + (−1.50 − 3.29i)6-s + (−0.809 − 0.934i)7-s + (−0.318 − 2.21i)8-s + (−1.91 − 0.563i)9-s + (3.42 − 3.95i)10-s + (0.642 + 0.413i)11-s + (1.16 + 0.747i)12-s + (−1.96 + 2.26i)13-s + (1.91 + 0.563i)14-s + (−1.02 − 7.16i)15-s + (3.17 + 3.66i)16-s + (−2.17 − 4.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.962 + 0.618i)2-s + (−0.183 + 1.27i)3-s + (0.128 − 0.281i)4-s + (−1.38 + 0.407i)5-s + (−0.613 − 1.34i)6-s + (−0.305 − 0.353i)7-s + (−0.112 − 0.782i)8-s + (−0.639 − 0.187i)9-s + (1.08 − 1.25i)10-s + (0.193 + 0.124i)11-s + (0.335 + 0.215i)12-s + (−0.544 + 0.628i)13-s + (0.512 + 0.150i)14-s + (−0.265 − 1.84i)15-s + (0.794 + 0.917i)16-s + (−0.527 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.119830 - 0.0193643i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.119830 - 0.0193643i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + (1.36 - 0.874i)T + (0.830 - 1.81i)T^{2} \) |
| 3 | \( 1 + (0.318 - 2.21i)T + (-2.87 - 0.845i)T^{2} \) |
| 5 | \( 1 + (3.10 - 0.911i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.934i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.642 - 0.413i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (1.96 - 2.26i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.17 + 4.76i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.830 + 1.81i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.24 + 2.72i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.954 - 6.63i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-3.10 - 0.911i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (5.25 - 1.54i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 + (5.54 + 6.40i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.61 + 1.86i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.55 - 10.8i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-6.08 + 3.91i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-6.53 + 4.19i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-6.42 + 14.0i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-4.54 + 5.24i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (12.6 + 3.72i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.217 - 1.51i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-4.11 + 1.20i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.58279080242597994753047651346, −9.716490496962338885650606720345, −9.165598445029620019738460066413, −8.170583225932989155883286057196, −7.20512945025646418566132199327, −6.73745253257475961314655198229, −4.90958371542832546087963374434, −4.10375575829004326273673841691, −3.24889845370958071053278714473, −0.12042494480980362735282747121,
1.10695804844631226782539337448, 2.44128679238057541126246392817, 3.94453055017619351120318564812, 5.44564715829317642217988127472, 6.56411824772624689113401841313, 7.78811697590044132756612176024, 8.048140064844036082286082262465, 8.988068187837834822458070296739, 10.01703933152537157356871158063, 11.09885159742488344523016318139
