| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 2·7-s + 8-s + 9-s − 10-s + 3·11-s + 12-s − 13-s − 2·14-s − 15-s + 16-s + 17-s + 18-s − 5·19-s − 20-s − 2·21-s + 3·22-s + 24-s − 4·25-s − 26-s + 27-s − 2·28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.14·19-s − 0.223·20-s − 0.436·21-s + 0.639·22-s + 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s − 0.377·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 911 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.29142998521495, −14.06520217116013, −13.27787561507099, −12.89165024661065, −12.53947739911814, −11.95350109888804, −11.49058873905332, −10.99899557282701, −10.25165790144940, −9.875816812496074, −9.325437233162890, −8.680830202724925, −8.328264346828646, −7.477382604857904, −7.209390983787357, −6.543616247236192, −6.099402818033297, −5.504998143375127, −4.723875011157804, −4.048283903315789, −3.800312654603967, −3.236254661044914, −2.425837883255415, −1.972688214647086, −1.019028974356116, 0,
1.019028974356116, 1.972688214647086, 2.425837883255415, 3.236254661044914, 3.800312654603967, 4.048283903315789, 4.723875011157804, 5.504998143375127, 6.099402818033297, 6.543616247236192, 7.209390983787357, 7.477382604857904, 8.328264346828646, 8.680830202724925, 9.325437233162890, 9.875816812496074, 10.25165790144940, 10.99899557282701, 11.49058873905332, 11.95350109888804, 12.53947739911814, 12.89165024661065, 13.27787561507099, 14.06520217116013, 14.29142998521495
