| L(s) = 1 | + (−0.309 + 0.951i)2-s + (0.978 + 0.207i)3-s + (−0.809 − 0.587i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−2.37 + 1.05i)7-s + (0.809 − 0.587i)8-s + (0.913 + 0.406i)9-s + (0.978 − 0.207i)10-s + (0.353 − 3.36i)11-s + (−0.669 − 0.743i)12-s + (−2.37 + 2.64i)13-s + (−0.271 − 2.58i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (0.621 + 5.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.564 + 0.120i)3-s + (−0.404 − 0.293i)4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (−0.897 + 0.399i)7-s + (0.286 − 0.207i)8-s + (0.304 + 0.135i)9-s + (0.309 − 0.0657i)10-s + (0.106 − 1.01i)11-s + (−0.193 − 0.214i)12-s + (−0.659 + 0.732i)13-s + (−0.0726 − 0.691i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (0.150 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.148546 + 0.789106i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.148546 + 0.789106i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 + (-0.978 - 0.207i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (5.19 + 2.01i)T \) |
| good | 7 | \( 1 + (2.37 - 1.05i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.353 + 3.36i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (2.37 - 2.64i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.621 - 5.91i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-2.86 - 3.17i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (6.68 - 4.85i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.88 - 5.79i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (1.62 - 2.81i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.35 + 1.77i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (5.24 + 5.82i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-0.947 - 2.91i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.49 + 1.55i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-11.5 - 2.45i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 + (-2.75 - 4.76i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.70 - 4.32i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (-0.0813 + 0.773i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (0.124 + 1.18i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (3.11 - 0.661i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-2.63 - 1.91i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (11.1 + 8.07i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.04158815966961427951440968238, −9.454268145160319392968339505457, −8.722792849967491819012734032676, −8.014048251134745691675016741681, −7.17552020261464620968990193587, −6.07151479047890974234775045264, −5.50026196812119032903706418171, −4.04804194234602476667557050732, −3.37413281948305217648900657411, −1.71644570179068053295066826545,
0.36682954232621124435918509151, 2.26984331698489244079072486504, 3.03001290692437230873854060062, 4.03261587389431402957660543880, 5.04691789758229208550110528639, 6.51945174264057639012371626838, 7.39421920715214824860601161632, 7.87799209922162020033308724781, 9.241094556305357074886148968264, 9.720793712546732239788383441605
