| L(s) = 1 | + 5.40·2-s − 2.97·3-s + 21.2·4-s − 3.19·5-s − 16.0·6-s − 30.2·7-s + 71.5·8-s − 18.1·9-s − 17.2·10-s − 6.23·11-s − 63.1·12-s − 33.6·13-s − 163.·14-s + 9.51·15-s + 216.·16-s − 76.1·17-s − 98.0·18-s − 12.1·19-s − 67.8·20-s + 90.0·21-s − 33.6·22-s − 212.·24-s − 114.·25-s − 181.·26-s + 134.·27-s − 642.·28-s + 83.7·29-s + ⋯ |
| L(s) = 1 | + 1.91·2-s − 0.572·3-s + 2.65·4-s − 0.285·5-s − 1.09·6-s − 1.63·7-s + 3.16·8-s − 0.672·9-s − 0.546·10-s − 0.170·11-s − 1.51·12-s − 0.717·13-s − 3.12·14-s + 0.163·15-s + 3.38·16-s − 1.08·17-s − 1.28·18-s − 0.147·19-s − 0.758·20-s + 0.935·21-s − 0.326·22-s − 1.80·24-s − 0.918·25-s − 1.37·26-s + 0.957·27-s − 4.33·28-s + 0.536·29-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 | \( 1 - 5.40T + 8T^{2} \) |
| 3 | \( 1 + 2.97T + 27T^{2} \) |
| 5 | \( 1 + 3.19T + 125T^{2} \) |
| 7 | \( 1 + 30.2T + 343T^{2} \) |
| 11 | \( 1 + 6.23T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 76.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 12.1T + 6.85e3T^{2} \) |
| 29 | \( 1 - 83.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 90.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 19.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 62.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 220.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 257.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 8.95T + 1.48e5T^{2} \) |
| 59 | \( 1 + 256.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 780.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 433.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 102.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 561.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 815.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 872.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 624.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 729.T + 9.12e5T^{2} \) |
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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
−10.48014327637613208206684414275, −9.304574031391692459878224719780, −7.69610563670479637821512530839, −6.63119421482753455340764598702, −6.20616447616667078209388793772, −5.27413128507894656877501986213, −4.27002896367916582218893121084, −3.25085101360824370336359823819, −2.41164180924467766867471100003, 0,
2.41164180924467766867471100003, 3.25085101360824370336359823819, 4.27002896367916582218893121084, 5.27413128507894656877501986213, 6.20616447616667078209388793772, 6.63119421482753455340764598702, 7.69610563670479637821512530839, 9.304574031391692459878224719780, 10.48014327637613208206684414275
