| L(s) = 1 | + (2.50 + 0.263i)2-s + (0.213 − 1.00i)3-s + (4.27 + 0.908i)4-s + (−2.23 + 0.125i)5-s + (0.800 − 2.46i)6-s + (−2.48 − 0.904i)7-s + (5.68 + 1.84i)8-s + (1.77 + 0.791i)9-s + (−5.63 − 0.272i)10-s + (−4.17 + 1.85i)11-s + (1.82 − 4.09i)12-s + (1.38 + 1.91i)13-s + (−6.00 − 2.92i)14-s + (−0.350 + 2.26i)15-s + (5.80 + 2.58i)16-s + (2.79 − 2.51i)17-s + ⋯ |
| L(s) = 1 | + (1.77 + 0.186i)2-s + (0.123 − 0.579i)3-s + (2.13 + 0.454i)4-s + (−0.998 + 0.0563i)5-s + (0.326 − 1.00i)6-s + (−0.939 − 0.341i)7-s + (2.01 + 0.653i)8-s + (0.592 + 0.263i)9-s + (−1.78 − 0.0863i)10-s + (−1.25 + 0.559i)11-s + (0.526 − 1.18i)12-s + (0.384 + 0.529i)13-s + (−1.60 − 0.781i)14-s + (−0.0904 + 0.585i)15-s + (1.45 + 0.645i)16-s + (0.677 − 0.610i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.49341 - 0.138573i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.49341 - 0.138573i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.23 - 0.125i)T \) |
| 7 | \( 1 + (2.48 + 0.904i)T \) |
| good | 2 | \( 1 + (-2.50 - 0.263i)T + (1.95 + 0.415i)T^{2} \) |
| 3 | \( 1 + (-0.213 + 1.00i)T + (-2.74 - 1.22i)T^{2} \) |
| 11 | \( 1 + (4.17 - 1.85i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-1.38 - 1.91i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.79 + 2.51i)T + (1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (4.00 - 0.851i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-1.49 - 0.157i)T + (22.4 + 4.78i)T^{2} \) |
| 29 | \( 1 + (2.65 + 8.17i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.00 - 4.45i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-2.49 + 5.60i)T + (-24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (3.92 - 2.85i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 7.33iT - 43T^{2} \) |
| 47 | \( 1 + (2.31 + 2.08i)T + (4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.504 + 2.37i)T + (-48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.482 - 4.58i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-1.10 + 10.5i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-0.411 + 0.370i)T + (7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (0.906 + 2.79i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.32 - 9.70i)T + (-48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (9.94 - 11.0i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-3.54 - 1.15i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.368 + 3.51i)T + (-87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (-12.8 + 4.19i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.88849366225037392534393621652, −12.21703135070576217419296718868, −11.17924900066673570023923023985, −10.04361825708443225129630451403, −7.957559661806161571376685096313, −7.19749728983687157163382594373, −6.36500207270670792572528405095, −4.85562638556887790472461229664, −3.88710969505378482845638394389, −2.61132654441104107305643047789,
3.02336482351644085353384987561, 3.71611978312973705721319547219, 4.88922771174616042420041242767, 5.99189487658825969760354138510, 7.17775096452309896179275017642, 8.563729575292612233753325763026, 10.27561711720644064414920022945, 10.90674778909110391394791920713, 12.14856898823229574105023175376, 12.82998173093910577742910550015
