| L(s) = 1 | + (−0.256 + 0.0834i)2-s + (−1.58 − 2.17i)3-s + (−1.55 + 1.13i)4-s + (−2.05 + 0.872i)5-s + (0.588 + 0.427i)6-s + 2.44i·7-s + (0.623 − 0.858i)8-s + (−1.31 + 4.03i)9-s + (0.455 − 0.395i)10-s + (−1.38 − 4.27i)11-s + (4.93 + 1.60i)12-s + (5.44 + 1.76i)13-s + (−0.203 − 0.626i)14-s + (5.15 + 3.10i)15-s + (1.10 − 3.39i)16-s + (0.587 − 0.809i)17-s + ⋯ |
| L(s) = 1 | + (−0.181 + 0.0590i)2-s + (−0.913 − 1.25i)3-s + (−0.779 + 0.566i)4-s + (−0.920 + 0.390i)5-s + (0.240 + 0.174i)6-s + 0.922i·7-s + (0.220 − 0.303i)8-s + (−0.437 + 1.34i)9-s + (0.144 − 0.125i)10-s + (−0.418 − 1.28i)11-s + (1.42 + 0.462i)12-s + (1.51 + 0.490i)13-s + (−0.0544 − 0.167i)14-s + (1.33 + 0.801i)15-s + (0.275 − 0.848i)16-s + (0.142 − 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.581479 - 0.115428i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.581479 - 0.115428i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.05 - 0.872i)T \) |
| 17 | \( 1 + (-0.587 + 0.809i)T \) |
| good | 2 | \( 1 + (0.256 - 0.0834i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (1.58 + 2.17i)T + (-0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 + (1.38 + 4.27i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-5.44 - 1.76i)T + (10.5 + 7.64i)T^{2} \) |
| 19 | \( 1 + (-0.0243 - 0.0177i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (4.77 - 1.54i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.83 + 4.24i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.19 - 3.04i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-7.37 - 2.39i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.945 + 2.90i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 4.13iT - 43T^{2} \) |
| 47 | \( 1 + (-6.38 - 8.78i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.56 - 6.28i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.13 + 3.50i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.443 + 1.36i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (4.13 - 5.69i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (8.22 - 5.97i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.95 + 1.28i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.01 - 0.738i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-7.90 + 10.8i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (3.20 + 9.85i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (5.67 + 7.80i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.57026098308710716497530640139, −10.50166431266007114659647496961, −8.867366113455973247867512024501, −8.266940812255869654954105041271, −7.56494628805762371579721421893, −6.33982394780776075678583529816, −5.73254451394572747153671087396, −4.21066057240664515858961137629, −2.91752251349347109369372390981, −0.78682210621406714606894861116,
0.805200085060670740991784076505, 3.87370603157543382707359846271, 4.34401952955050649119996223706, 5.16367707604761573009857024828, 6.26218764291724268783231727728, 7.75000506143660685549918308842, 8.646545037733534776360428906722, 9.755218198776124775178967340069, 10.40038986925347788384938822063, 10.86512784815721866437052464856
