| L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.913 − 0.406i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (2.56 − 2.85i)7-s + (−0.309 + 0.951i)8-s + (0.669 + 0.743i)9-s + (−0.913 + 0.406i)10-s + (−1.87 − 0.399i)11-s + (0.104 − 0.994i)12-s + (0.00217 + 0.0206i)13-s + (3.75 − 0.798i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (7.11 − 1.51i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.527 − 0.234i)3-s + (0.154 + 0.475i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (0.971 − 1.07i)7-s + (−0.109 + 0.336i)8-s + (0.223 + 0.247i)9-s + (−0.288 + 0.128i)10-s + (−0.566 − 0.120i)11-s + (0.0301 − 0.287i)12-s + (0.000603 + 0.00573i)13-s + (1.00 − 0.213i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (1.72 − 0.366i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0264i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.96442 + 0.0259491i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96442 + 0.0259491i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (0.913 + 0.406i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-1.01 - 5.47i)T \) |
| good | 7 | \( 1 + (-2.56 + 2.85i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (1.87 + 0.399i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.00217 - 0.0206i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (-7.11 + 1.51i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.814 + 7.74i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (1.30 - 4.03i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-3.86 - 2.80i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (0.186 + 0.322i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-10.7 + 4.76i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.832 + 7.91i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-1.38 + 1.00i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-8.49 - 9.43i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (7.66 + 3.41i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 + (2.58 - 4.48i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.51 + 1.68i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-2.79 - 0.594i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-14.0 + 2.99i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (0.768 - 0.342i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (0.939 + 2.89i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.22 - 3.76i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.53940831138413359982526345779, −9.169881471472146860772003179008, −7.899117247657551826406257043427, −7.45652748257699722416212276666, −6.81701306902776270896947148314, −5.53435009298966430318459998218, −4.95725016353964887353697392819, −3.94055720757182118240136784578, −2.77872369096445973060582249599, −1.01229598707864406955878229172,
1.29765616954119386453417885353, 2.57398658945034325949862832535, 3.90775386855745235288673156633, 4.81783485807973873826241556172, 5.63998375068265692823202569336, 6.09664321728803942540348396984, 7.87202433692421331037095887616, 8.135469290354270445672289326279, 9.495980701375694000027801026473, 10.17684985668366432482237116206
