| L(s) = 1 | + (0.173 − 0.984i)2-s + (1.05 + 0.883i)3-s + (−0.939 − 0.342i)4-s + (2.40 − 0.876i)5-s + (1.05 − 0.883i)6-s + (2.52 + 4.37i)7-s + (−0.5 + 0.866i)8-s + (−0.192 − 1.09i)9-s + (−0.444 − 2.52i)10-s + (0.5 − 0.866i)11-s + (−0.687 − 1.19i)12-s + (−5.16 + 4.33i)13-s + (4.74 − 1.72i)14-s + (3.30 + 1.20i)15-s + (0.766 + 0.642i)16-s + (0.720 − 4.08i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.608 + 0.510i)3-s + (−0.469 − 0.171i)4-s + (1.07 − 0.391i)5-s + (0.430 − 0.360i)6-s + (0.954 + 1.65i)7-s + (−0.176 + 0.306i)8-s + (−0.0641 − 0.364i)9-s + (−0.140 − 0.797i)10-s + (0.150 − 0.261i)11-s + (−0.198 − 0.343i)12-s + (−1.43 + 1.20i)13-s + (1.26 − 0.461i)14-s + (0.854 + 0.311i)15-s + (0.191 + 0.160i)16-s + (0.174 − 0.991i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.04883 - 0.217627i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04883 - 0.217627i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-1.40 - 4.12i)T \) |
| good | 3 | \( 1 + (-1.05 - 0.883i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-2.40 + 0.876i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-2.52 - 4.37i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (5.16 - 4.33i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.720 + 4.08i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.440 - 0.160i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.29 + 7.34i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (4.62 + 8.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.31T + 37T^{2} \) |
| 41 | \( 1 + (-0.554 - 0.465i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-6.08 + 2.21i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.419 + 2.38i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (8.29 + 3.01i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.906 + 5.14i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.511 - 0.186i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.31 - 7.48i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (12.1 - 4.40i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-2.39 - 2.00i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (6.70 + 5.62i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.776 - 1.34i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.18 + 4.35i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.466 + 2.64i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.55859407454644680624682919558, −9.811187576855092833144443433513, −9.453557802603464040181878820125, −8.949967628673748158042680115858, −7.81543321618378931797099660339, −6.01677133937142184258804311839, −5.27953557405863679913141009612, −4.26686733396008089028580935465, −2.62133129988056904292758519115, −1.95586414195547656640666155193,
1.54935064985269166699598824344, 3.01136711937931388439822201872, 4.62383701656794967331287484625, 5.44163975811529500751877611162, 6.91365509531054786000132582666, 7.43753871900085873067693177492, 8.122965360353585648403171519210, 9.346244088074204707946455866449, 10.44247558523765719586548494673, 10.79593956717839558929420588435
