| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 25·5-s − 36·6-s − 27.1·7-s + 64·8-s + 81·9-s + 100·10-s + 375.·11-s − 144·12-s − 531.·13-s − 108.·14-s − 225·15-s + 256·16-s − 1.08e3·17-s + 324·18-s − 361·19-s + 400·20-s + 244.·21-s + 1.50e3·22-s − 3.98e3·23-s − 576·24-s + 625·25-s − 2.12e3·26-s − 729·27-s − 434.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.209·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.935·11-s − 0.288·12-s − 0.872·13-s − 0.148·14-s − 0.258·15-s + 0.250·16-s − 0.909·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.120·21-s + 0.661·22-s − 1.57·23-s − 0.204·24-s + 0.200·25-s − 0.617·26-s − 0.192·27-s − 0.104·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + 9T \) |
| 5 | \( 1 - 25T \) |
| 19 | \( 1 + 361T \) |
| good | 7 | \( 1 + 27.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 375.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 531.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.08e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.98e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 87.7T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.32e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.81e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.51e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.41e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.79e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.14e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.38e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.25e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.85e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.71e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.34e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.23e5T + 8.58e9T^{2} \) |
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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
−9.773600022820759580950354507987, −8.685225024500311907603085864290, −7.41742102861829972482881092776, −6.48732910378817295827238153037, −5.94910934925261142627198894931, −4.77029830410098606519998148546, −4.04376112828898416230801434309, −2.62630847592069073350748406100, −1.53853954107180083759581549896, 0,
1.53853954107180083759581549896, 2.62630847592069073350748406100, 4.04376112828898416230801434309, 4.77029830410098606519998148546, 5.94910934925261142627198894931, 6.48732910378817295827238153037, 7.41742102861829972482881092776, 8.685225024500311907603085864290, 9.773600022820759580950354507987
