| L(s) = 1 | − 0.535·2-s − 3·3-s − 7.71·4-s + 12.7·5-s + 1.60·6-s − 8.67·7-s + 8.41·8-s + 9·9-s − 6.80·10-s + 52.9·11-s + 23.1·12-s − 73.3·13-s + 4.64·14-s − 38.1·15-s + 57.2·16-s − 23.1·17-s − 4.81·18-s − 91.6·19-s − 97.9·20-s + 26.0·21-s − 28.3·22-s + 7.43·23-s − 25.2·24-s + 36.3·25-s + 39.2·26-s − 27·27-s + 66.9·28-s + ⋯ |
| L(s) = 1 | − 0.189·2-s − 0.577·3-s − 0.964·4-s + 1.13·5-s + 0.109·6-s − 0.468·7-s + 0.371·8-s + 0.333·9-s − 0.215·10-s + 1.45·11-s + 0.556·12-s − 1.56·13-s + 0.0886·14-s − 0.656·15-s + 0.893·16-s − 0.330·17-s − 0.0630·18-s − 1.10·19-s − 1.09·20-s + 0.270·21-s − 0.274·22-s + 0.0674·23-s − 0.214·24-s + 0.291·25-s + 0.296·26-s − 0.192·27-s + 0.451·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 67 | \( 1 - 67T \) |
| good | 2 | \( 1 + 0.535T + 8T^{2} \) |
| 5 | \( 1 - 12.7T + 125T^{2} \) |
| 7 | \( 1 + 8.67T + 343T^{2} \) |
| 11 | \( 1 - 52.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 73.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 23.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 91.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 7.43T + 1.21e4T^{2} \) |
| 29 | \( 1 + 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 304.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 227.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 169.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 398.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 327.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 21.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 76.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 639.T + 2.26e5T^{2} \) |
| 71 | \( 1 + 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 822.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 883.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 967.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.14e3T + 9.12e5T^{2} \) |
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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
−11.52346256883406774001399490793, −10.20337345885642568838247815733, −9.556429807758256600442225595980, −8.919170637413611415938693367266, −7.24218047969337524860256800676, −6.16564977061532937386079742212, −5.15408313951818526000535237729, −3.95680369860517672465497797938, −1.84421275672066076522139511220, 0,
1.84421275672066076522139511220, 3.95680369860517672465497797938, 5.15408313951818526000535237729, 6.16564977061532937386079742212, 7.24218047969337524860256800676, 8.919170637413611415938693367266, 9.556429807758256600442225595980, 10.20337345885642568838247815733, 11.52346256883406774001399490793
