L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 25·5-s + 36·6-s + 241.·7-s − 64·8-s + 81·9-s + 100·10-s + 48.9·11-s − 144·12-s − 273.·13-s − 966.·14-s + 225·15-s + 256·16-s + 601.·17-s − 324·18-s − 361·19-s − 400·20-s − 2.17e3·21-s − 195.·22-s − 4.65e3·23-s + 576·24-s + 625·25-s + 1.09e3·26-s − 729·27-s + 3.86e3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.86·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.121·11-s − 0.288·12-s − 0.449·13-s − 1.31·14-s + 0.258·15-s + 0.250·16-s + 0.504·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 1.07·21-s − 0.0861·22-s − 1.83·23-s + 0.204·24-s + 0.200·25-s + 0.317·26-s − 0.192·27-s + 0.932·28-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 | \( 1 - 241.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 48.9T + 1.61e5T^{2} \) |
| 13 | \( 1 + 273.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 601.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.65e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.70e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.34e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.24e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.46e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.59e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.29e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.88e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.70e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.81e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.78e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.76e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.51e4T + 8.58e9T^{2} \) |
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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
−9.576494390803910021278616254597, −8.377901455264556733991889725383, −7.900104997601368299315358711282, −7.07434157365350254043848693448, −5.81559185218754669822201914365, −4.91307424914966841983557592195, −3.93174763771776075822175754702, −2.16601137572792813473272458954, −1.26458063222869887435969824461, 0,
1.26458063222869887435969824461, 2.16601137572792813473272458954, 3.93174763771776075822175754702, 4.91307424914966841983557592195, 5.81559185218754669822201914365, 7.07434157365350254043848693448, 7.900104997601368299315358711282, 8.377901455264556733991889725383, 9.576494390803910021278616254597