| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 16.3·5-s − 6·6-s − 22.6·7-s − 8·8-s + 9·9-s − 32.6·10-s + 3.51·11-s + 12·12-s − 40.6·13-s + 45.2·14-s + 48.9·15-s + 16·16-s − 39.8·17-s − 18·18-s + 65.2·20-s − 67.8·21-s − 7.02·22-s + 11.0·23-s − 24·24-s + 140.·25-s + 81.3·26-s + 27·27-s − 90.4·28-s + 240.·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.45·5-s − 0.408·6-s − 1.22·7-s − 0.353·8-s + 0.333·9-s − 1.03·10-s + 0.0962·11-s + 0.288·12-s − 0.867·13-s + 0.863·14-s + 0.841·15-s + 0.250·16-s − 0.569·17-s − 0.235·18-s + 0.729·20-s − 0.704·21-s − 0.0680·22-s + 0.100·23-s − 0.204·24-s + 1.12·25-s + 0.613·26-s + 0.192·27-s − 0.610·28-s + 1.54·29-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 | \( 1 - 16.3T + 125T^{2} \) |
| 7 | \( 1 + 22.6T + 343T^{2} \) |
| 11 | \( 1 - 3.51T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.8T + 4.91e3T^{2} \) |
| 23 | \( 1 - 11.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 240.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 391.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 290.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 506.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 356.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 274.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 179.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 527.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 162.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 514.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 404.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 730.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 37.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.64e3T + 9.12e5T^{2} \) |
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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
−8.752751249133761716418116501880, −7.43163107350050664593829454089, −6.89715461016623629062645404293, −6.11541501797757338963856854391, −5.40542485995818820988879754405, −4.12316169327479697607532475846, −2.85686857582902961188776399499, −2.41141786170302693699737821435, −1.34695472932354955501186658471, 0,
1.34695472932354955501186658471, 2.41141786170302693699737821435, 2.85686857582902961188776399499, 4.12316169327479697607532475846, 5.40542485995818820988879754405, 6.11541501797757338963856854391, 6.89715461016623629062645404293, 7.43163107350050664593829454089, 8.752751249133761716418116501880
