L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 25·5-s + 36·6-s − 198.·7-s − 64·8-s + 81·9-s + 100·10-s − 301.·11-s − 144·12-s − 414.·13-s + 794.·14-s + 225·15-s + 256·16-s + 1.80e3·17-s − 324·18-s − 361·19-s − 400·20-s + 1.78e3·21-s + 1.20e3·22-s + 2.57e3·23-s + 576·24-s + 625·25-s + 1.65e3·26-s − 729·27-s − 3.17e3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 1.53·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.751·11-s − 0.288·12-s − 0.679·13-s + 1.08·14-s + 0.258·15-s + 0.250·16-s + 1.51·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.885·21-s + 0.531·22-s + 1.01·23-s + 0.204·24-s + 0.200·25-s + 0.480·26-s − 0.192·27-s − 0.766·28-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 | \( 1 + 198.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 301.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 414.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.80e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.84e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 510.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.45e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.55e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.11e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.19e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.99e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.28e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.54e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e5T + 8.58e9T^{2} \) |
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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
−9.822159373374451461049428422137, −8.698330169684823693338265899400, −7.60892086288757079581576678050, −6.95038657003505887248829831724, −6.00859339196215832442656278389, −5.02793325517120215117032622768, −3.52454553963531403643495203832, −2.65571612103618992643526952730, −0.895526096202802781658008331652, 0,
0.895526096202802781658008331652, 2.65571612103618992643526952730, 3.52454553963531403643495203832, 5.02793325517120215117032622768, 6.00859339196215832442656278389, 6.95038657003505887248829831724, 7.60892086288757079581576678050, 8.698330169684823693338265899400, 9.822159373374451461049428422137