| L(s) = 1 | + 4.24·2-s − 4.41·3-s + 9.98·4-s − 7.80·5-s − 18.7·6-s + 27.0·7-s + 8.43·8-s − 7.48·9-s − 33.1·10-s − 35.2·11-s − 44.1·12-s + 58.1·13-s + 114.·14-s + 34.4·15-s − 44.1·16-s − 98.3·17-s − 31.7·18-s + 35.3·19-s − 77.9·20-s − 119.·21-s − 149.·22-s − 37.2·24-s − 64.0·25-s + 246.·26-s + 152.·27-s + 270.·28-s − 235.·29-s + ⋯ |
| L(s) = 1 | + 1.49·2-s − 0.850·3-s + 1.24·4-s − 0.698·5-s − 1.27·6-s + 1.46·7-s + 0.372·8-s − 0.277·9-s − 1.04·10-s − 0.966·11-s − 1.06·12-s + 1.24·13-s + 2.19·14-s + 0.593·15-s − 0.689·16-s − 1.40·17-s − 0.415·18-s + 0.426·19-s − 0.871·20-s − 1.24·21-s − 1.44·22-s − 0.317·24-s − 0.512·25-s + 1.86·26-s + 1.08·27-s + 1.82·28-s − 1.50·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 \) |
| good | 2 | \( 1 - 4.24T + 8T^{2} \) |
| 3 | \( 1 + 4.41T + 27T^{2} \) |
| 5 | \( 1 + 7.80T + 125T^{2} \) |
| 7 | \( 1 - 27.0T + 343T^{2} \) |
| 11 | \( 1 + 35.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 98.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 35.3T + 6.85e3T^{2} \) |
| 29 | \( 1 + 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 55.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 401.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 59.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 11.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 103.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 351.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 547.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 478.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 14.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 843.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 118.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 388.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 62.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + 678.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 421.T + 9.12e5T^{2} \) |
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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
−10.89001510101571674679383907928, −8.874765005709579033381631886180, −8.043221151327483623943854429201, −6.96590812517798797361763826880, −5.83051214560024905598016171348, −5.22092004566653856494916960005, −4.44734524408175940803082472618, −3.45878775380885906415778403098, −1.95339051485587450227408974080, 0,
1.95339051485587450227408974080, 3.45878775380885906415778403098, 4.44734524408175940803082472618, 5.22092004566653856494916960005, 5.83051214560024905598016171348, 6.96590812517798797361763826880, 8.043221151327483623943854429201, 8.874765005709579033381631886180, 10.89001510101571674679383907928
