| L(s) = 1 | + 1.17·2-s + 9.69·3-s − 6.62·4-s − 0.587·5-s + 11.3·6-s − 24.4·7-s − 17.1·8-s + 66.9·9-s − 0.689·10-s − 30.6·11-s − 64.1·12-s − 27.6·13-s − 28.7·14-s − 5.69·15-s + 32.8·16-s + 30.7·17-s + 78.6·18-s − 92.5·19-s + 3.88·20-s − 237.·21-s − 35.9·22-s − 166.·24-s − 124.·25-s − 32.4·26-s + 387.·27-s + 161.·28-s − 73.6·29-s + ⋯ |
| L(s) = 1 | + 0.415·2-s + 1.86·3-s − 0.827·4-s − 0.0525·5-s + 0.774·6-s − 1.32·7-s − 0.758·8-s + 2.48·9-s − 0.0218·10-s − 0.840·11-s − 1.54·12-s − 0.589·13-s − 0.548·14-s − 0.0980·15-s + 0.512·16-s + 0.439·17-s + 1.02·18-s − 1.11·19-s + 0.0434·20-s − 2.46·21-s − 0.348·22-s − 1.41·24-s − 0.997·25-s − 0.244·26-s + 2.76·27-s + 1.09·28-s − 0.471·29-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 | \( 1 - 1.17T + 8T^{2} \) |
| 3 | \( 1 - 9.69T + 27T^{2} \) |
| 5 | \( 1 + 0.587T + 125T^{2} \) |
| 7 | \( 1 + 24.4T + 343T^{2} \) |
| 11 | \( 1 + 30.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 27.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 30.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92.5T + 6.85e3T^{2} \) |
| 29 | \( 1 + 73.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 105.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 89.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 88.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 181.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 612.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 78.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 111.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 408.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 449.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 93.4T + 3.89e5T^{2} \) |
| 79 | \( 1 - 277.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.27e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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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
−9.719369203717683228795142583402, −9.179523110337070782250775602103, −8.283488754278951983978950287657, −7.56553605140577245120299409979, −6.38918307496856830006457599668, −4.99028376944369475159643050680, −3.80990577920215339492792685912, −3.25411294296517964846426464118, −2.19048974265544678135784786506, 0,
2.19048974265544678135784786506, 3.25411294296517964846426464118, 3.80990577920215339492792685912, 4.99028376944369475159643050680, 6.38918307496856830006457599668, 7.56553605140577245120299409979, 8.283488754278951983978950287657, 9.179523110337070782250775602103, 9.719369203717683228795142583402
