| L(s) = 1 | + (0.286 + 1.38i)2-s + (−0.360 − 0.469i)3-s + (−1.83 + 0.794i)4-s + (2.24 − 2.92i)5-s + (0.546 − 0.633i)6-s + (2.53 + 0.750i)7-s + (−1.62 − 2.31i)8-s + (0.685 − 2.55i)9-s + (4.69 + 2.26i)10-s + (0.0254 − 0.193i)11-s + (1.03 + 0.575i)12-s + (−1.79 + 4.33i)13-s + (−0.311 + 3.72i)14-s − 2.18·15-s + (2.73 − 2.91i)16-s + (1.62 − 2.82i)17-s + ⋯ |
| L(s) = 1 | + (0.202 + 0.979i)2-s + (−0.208 − 0.271i)3-s + (−0.917 + 0.397i)4-s + (1.00 − 1.30i)5-s + (0.223 − 0.258i)6-s + (0.958 + 0.283i)7-s + (−0.575 − 0.818i)8-s + (0.228 − 0.853i)9-s + (1.48 + 0.717i)10-s + (0.00768 − 0.0583i)11-s + (0.298 + 0.166i)12-s + (−0.497 + 1.20i)13-s + (−0.0832 + 0.996i)14-s − 0.563·15-s + (0.684 − 0.729i)16-s + (0.395 − 0.684i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39167 + 0.283844i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39167 + 0.283844i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.286 - 1.38i)T \) |
| 7 | \( 1 + (-2.53 - 0.750i)T \) |
| good | 3 | \( 1 + (0.360 + 0.469i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (-2.24 + 2.92i)T + (-1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (-0.0254 + 0.193i)T + (-10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (1.79 - 4.33i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (-1.62 + 2.82i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.981 - 7.45i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (0.849 + 0.227i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.793 - 1.91i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (-1.85 + 3.20i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (8.19 + 6.28i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (3.81 - 3.81i)T - 41iT^{2} \) |
| 43 | \( 1 + (2.28 - 0.945i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (1.58 - 0.913i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.93 - 1.17i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (1.11 - 8.45i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (0.828 + 6.29i)T + (-58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (9.50 - 7.29i)T + (17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (4.95 + 4.95i)T + 71iT^{2} \) |
| 73 | \( 1 + (-2.03 - 7.60i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.39 - 4.14i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.02 + 2.08i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-0.0259 + 0.0968i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 - 9.39iT - 97T^{2} \) |
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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
−12.21812890990154265410599961405, −12.01779210789734979539191730965, −9.904537567931661484686706488433, −9.193062261945043594508326243845, −8.388161004868766097808806573680, −7.20809872396066957165499421022, −5.95765279284234498330803534143, −5.24747684429537398207243590126, −4.17692387881671038099436495046, −1.50642592018329373440134416793,
1.95919050328576129388114398998, 3.10850698049288749792078951604, 4.80813525156624837539642326569, 5.60055692762656036800401209493, 7.13559567210583507356572387271, 8.380550147213910362532980380759, 9.856700122126297050209053070689, 10.50111499694547781233865457129, 10.89098986573458099117094971724, 11.96892506535834239158676744731
