| L(s) = 1 | + 3.48·2-s + 4.14·4-s − 7·7-s − 13.4·8-s + 6.90·11-s + 22.1·13-s − 24.3·14-s − 79.9·16-s + 88.3·17-s + 36.9·19-s + 24.0·22-s − 95.5·23-s + 77.1·26-s − 29.0·28-s − 269.·29-s + 197.·31-s − 171.·32-s + 307.·34-s − 2.14·37-s + 128.·38-s − 174.·41-s + 17.0·43-s + 28.6·44-s − 332.·46-s − 528.·47-s + 49·49-s + 91.8·52-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.518·4-s − 0.377·7-s − 0.593·8-s + 0.189·11-s + 0.472·13-s − 0.465·14-s − 1.24·16-s + 1.25·17-s + 0.446·19-s + 0.233·22-s − 0.866·23-s + 0.582·26-s − 0.196·28-s − 1.72·29-s + 1.14·31-s − 0.946·32-s + 1.55·34-s − 0.00953·37-s + 0.549·38-s − 0.663·41-s + 0.0604·43-s + 0.0982·44-s − 1.06·46-s − 1.63·47-s + 0.142·49-s + 0.244·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 3.48T + 8T^{2} \) |
| 11 | \( 1 - 6.90T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 88.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 36.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 95.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 269.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 2.14T + 5.06e4T^{2} \) |
| 41 | \( 1 + 174.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 17.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 528.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 641.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 642.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 142.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 478.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 105.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 986.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 711.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 636.T + 9.12e5T^{2} \) |
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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
−8.631274806910705058641459064485, −7.76428884452982676132235036260, −6.76909309130416818080287817997, −5.94717664091341767994862368557, −5.41454971338769538128776229066, −4.39327284002764716379220013273, −3.57628441840532455417717154157, −2.93852153155890857016084801952, −1.53257828215893173518559705300, 0,
1.53257828215893173518559705300, 2.93852153155890857016084801952, 3.57628441840532455417717154157, 4.39327284002764716379220013273, 5.41454971338769538128776229066, 5.94717664091341767994862368557, 6.76909309130416818080287817997, 7.76428884452982676132235036260, 8.631274806910705058641459064485
