L(s) = 1 | + (−2.24 + 1.29i)2-s − 0.518·3-s + (2.35 − 4.07i)4-s + (−1.39 − 0.806i)5-s + (1.16 − 0.671i)6-s + 6.99i·8-s − 2.73·9-s + 4.17·10-s + 2.70i·11-s + (−1.21 + 2.11i)12-s + (−2.36 − 2.71i)13-s + (0.723 + 0.417i)15-s + (−4.34 − 7.53i)16-s + (−1.56 + 2.70i)17-s + (6.12 − 3.53i)18-s + 3.68i·19-s + ⋯ |
L(s) = 1 | + (−1.58 + 0.915i)2-s − 0.299·3-s + (1.17 − 2.03i)4-s + (−0.624 − 0.360i)5-s + (0.474 − 0.273i)6-s + 2.47i·8-s − 0.910·9-s + 1.31·10-s + 0.815i·11-s + (−0.351 + 0.609i)12-s + (−0.656 − 0.753i)13-s + (0.186 + 0.107i)15-s + (−1.08 − 1.88i)16-s + (−0.379 + 0.656i)17-s + (1.44 − 0.833i)18-s + 0.844i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.376839 + 0.117136i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.376839 + 0.117136i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (2.36 + 2.71i)T \) |
good | 2 | \( 1 + (2.24 - 1.29i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + 0.518T + 3T^{2} \) |
| 5 | \( 1 + (1.39 + 0.806i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 2.70iT - 11T^{2} \) |
| 17 | \( 1 + (1.56 - 2.70i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 3.68iT - 19T^{2} \) |
| 23 | \( 1 + (-0.993 - 1.71i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.68 + 4.65i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-9.07 + 5.23i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.15 + 2.97i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.66 - 3.85i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.67 + 2.90i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.913 - 0.527i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.63 + 6.29i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.89 - 5.71i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 - 13.5iT - 67T^{2} \) |
| 71 | \( 1 + (-1.17 + 0.675i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.88 - 4.55i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.10 + 5.37i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.69iT - 83T^{2} \) |
| 89 | \( 1 + (1.52 - 0.879i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.4 + 7.74i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.20296647062042582024204021306, −9.831077271988308185465732287935, −8.652128252743173736343428585761, −8.074335587175804391405070188803, −7.48433149041329000339765440355, −6.32603022152793095531206010852, −5.65425382077585683946275527844, −4.40137338271819921864584958456, −2.39305808741371959166309843252, −0.63969520050714243277928791517,
0.70366555032453540690329072153, 2.53464438834220998502540587092, 3.24062612781280193779829139892, 4.77945050151903595906820342868, 6.44654024930490653198588451613, 7.22869814973935348263237056006, 8.196619112792557235691085669220, 8.863900997069441388085856935565, 9.574756559730280560449987438758, 10.67526396999688094871362802329