| L(s) = 1 | − 2.31·2-s + 8.67·3-s − 2.62·4-s − 5·5-s − 20.1·6-s − 21.6·7-s + 24.6·8-s + 48.3·9-s + 11.5·10-s + 27.9·11-s − 22.7·12-s − 13·13-s + 50.3·14-s − 43.3·15-s − 36.1·16-s − 63.4·17-s − 112.·18-s + 34.1·19-s + 13.1·20-s − 188.·21-s − 64.9·22-s − 1.61·23-s + 213.·24-s + 25·25-s + 30.1·26-s + 185.·27-s + 56.8·28-s + ⋯ |
| L(s) = 1 | − 0.820·2-s + 1.67·3-s − 0.327·4-s − 0.447·5-s − 1.36·6-s − 1.17·7-s + 1.08·8-s + 1.79·9-s + 0.366·10-s + 0.767·11-s − 0.547·12-s − 0.277·13-s + 0.960·14-s − 0.747·15-s − 0.565·16-s − 0.905·17-s − 1.46·18-s + 0.412·19-s + 0.146·20-s − 1.95·21-s − 0.629·22-s − 0.0146·23-s + 1.81·24-s + 0.200·25-s + 0.227·26-s + 1.32·27-s + 0.383·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 + 2.31T + 8T^{2} \) |
| 3 | \( 1 - 8.67T + 27T^{2} \) |
| 7 | \( 1 + 21.6T + 343T^{2} \) |
| 11 | \( 1 - 27.9T + 1.33e3T^{2} \) |
| 17 | \( 1 + 63.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.61T + 1.21e4T^{2} \) |
| 29 | \( 1 - 155.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 8.86T + 5.06e4T^{2} \) |
| 41 | \( 1 - 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 342.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 433.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 20.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 270.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 22.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 226.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 989.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 535.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 902.T + 9.12e5T^{2} \) |
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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
−8.512108033262246837559457462127, −7.974298268988476565533855078903, −7.07033543255747366757472866070, −6.52710378930347752532327415944, −4.85305614191162107961678612444, −3.98581033743246001709415703496, −3.34854538387951895507598651313, −2.40234919996679594601883770650, −1.23903232716149343525751683283, 0,
1.23903232716149343525751683283, 2.40234919996679594601883770650, 3.34854538387951895507598651313, 3.98581033743246001709415703496, 4.85305614191162107961678612444, 6.52710378930347752532327415944, 7.07033543255747366757472866070, 7.974298268988476565533855078903, 8.512108033262246837559457462127
