| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 6.58·5-s − 6·6-s − 12.5·7-s + 8·8-s + 9·9-s + 13.1·10-s − 47.3·11-s − 12·12-s + 79.8·13-s − 25.1·14-s − 19.7·15-s + 16·16-s + 80.7·17-s + 18·18-s + 26.3·20-s + 37.7·21-s − 94.7·22-s − 214.·23-s − 24·24-s − 81.5·25-s + 159.·26-s − 27·27-s − 50.3·28-s − 72.7·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.589·5-s − 0.408·6-s − 0.680·7-s + 0.353·8-s + 0.333·9-s + 0.416·10-s − 1.29·11-s − 0.288·12-s + 1.70·13-s − 0.480·14-s − 0.340·15-s + 0.250·16-s + 1.15·17-s + 0.235·18-s + 0.294·20-s + 0.392·21-s − 0.918·22-s − 1.94·23-s − 0.204·24-s − 0.652·25-s + 1.20·26-s − 0.192·27-s − 0.340·28-s − 0.465·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 6.58T + 125T^{2} \) |
| 7 | \( 1 + 12.5T + 343T^{2} \) |
| 11 | \( 1 + 47.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 79.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 80.7T + 4.91e3T^{2} \) |
| 23 | \( 1 + 214.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 72.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 1.68T + 5.06e4T^{2} \) |
| 41 | \( 1 - 483.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 350.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 137.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 97.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 830.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 30.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 333.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 444.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 198.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 389.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 216.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.39e3T + 9.12e5T^{2} \) |
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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
−8.045513902484482487868205619663, −7.55793255819446511921045722230, −6.34630967534352935574026456486, −5.76998154827602392432027395867, −5.55454089667403792250253006569, −4.17610070832057138296247275109, −3.51306881400470143671234953664, −2.42234985161791631835461315010, −1.36827118390069620084209205112, 0,
1.36827118390069620084209205112, 2.42234985161791631835461315010, 3.51306881400470143671234953664, 4.17610070832057138296247275109, 5.55454089667403792250253006569, 5.76998154827602392432027395867, 6.34630967534352935574026456486, 7.55793255819446511921045722230, 8.045513902484482487868205619663
