| L(s) = 1 | + (1.33 + 2.31i)2-s + (1.04 − 0.874i)3-s + (−2.58 + 4.47i)4-s + (−0.127 − 0.725i)5-s + (3.42 + 1.24i)6-s + (−0.876 − 4.97i)7-s − 8.46·8-s + (−0.199 + 1.13i)9-s + (1.50 − 1.26i)10-s + (−2.15 − 1.80i)11-s + (1.21 + 6.91i)12-s + (0.730 + 4.14i)13-s + (10.3 − 8.68i)14-s + (−0.767 − 0.643i)15-s + (−6.16 − 10.6i)16-s + (0.913 + 1.58i)17-s + ⋯ |
| L(s) = 1 | + (0.946 + 1.63i)2-s + (0.601 − 0.504i)3-s + (−1.29 + 2.23i)4-s + (−0.0571 − 0.324i)5-s + (1.39 + 0.508i)6-s + (−0.331 − 1.87i)7-s − 2.99·8-s + (−0.0665 + 0.377i)9-s + (0.477 − 0.400i)10-s + (−0.649 − 0.544i)11-s + (0.351 + 1.99i)12-s + (0.202 + 1.14i)13-s + (2.76 − 2.32i)14-s + (−0.198 − 0.166i)15-s + (−1.54 − 2.66i)16-s + (0.221 + 0.383i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0148 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0148 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15149 + 1.13453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15149 + 1.13453i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (1.62 + 10.3i)T \) |
| good | 2 | \( 1 + (-1.33 - 2.31i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.04 + 0.874i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (0.127 + 0.725i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.876 + 4.97i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (2.15 + 1.80i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.730 - 4.14i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.913 - 1.58i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.447 - 0.774i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.13 - 1.95i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.71 - 3.11i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.777 + 4.40i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.56 + 8.90i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 - 4.99T + 41T^{2} \) |
| 43 | \( 1 + (-1.71 - 2.97i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.27 - 1.19i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.734 - 4.16i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (4.15 + 3.48i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (7.23 + 2.63i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.498 + 2.82i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.87 + 3.24i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.58 + 3.00i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.33 + 7.59i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.769 + 0.645i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-6.29 - 2.29i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-2.36 - 13.4i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−13.97588378982927235789556277154, −13.35913030915852627431481536918, −12.70352260959349132167256756258, −10.89077132129487087399779092652, −9.100076827840360451347709292119, −7.83070273667150364791950787791, −7.40332065413259579668929138265, −6.22634855107363987965759174609, −4.67403079686689877187160213974, −3.58669281144610423129116948799,
2.58872026793997817995313372485, 3.21271272712283681467359468444, 4.93323810586188426153266611144, 5.96754313618676548265331828899, 8.593514363316431139295921480544, 9.538818573891619639112856314764, 10.32411011317065630603375541025, 11.57565273803803333936075852350, 12.41011718140537761442697137699, 13.06591892192840896000086273206
