L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 12.9·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s − 25.9·10-s + 48.3·11-s + 12·12-s − 39.8·13-s − 14·14-s − 38.8·15-s + 16·16-s − 64.1·17-s + 18·18-s + 19·19-s − 51.8·20-s − 21·21-s + 96.6·22-s − 195.·23-s + 24·24-s + 43.1·25-s − 79.7·26-s + 27·27-s − 28·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.15·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.820·10-s + 1.32·11-s + 0.288·12-s − 0.850·13-s − 0.267·14-s − 0.669·15-s + 0.250·16-s − 0.915·17-s + 0.235·18-s + 0.229·19-s − 0.579·20-s − 0.218·21-s + 0.936·22-s − 1.77·23-s + 0.204·24-s + 0.344·25-s − 0.601·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 19 | \( 1 - 19T \) |
good | 5 | \( 1 + 12.9T + 125T^{2} \) |
| 11 | \( 1 - 48.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 39.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 64.1T + 4.91e3T^{2} \) |
| 23 | \( 1 + 195.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 234.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 110.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 242.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 167.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 285.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 354.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 20.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 914.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 282.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 812.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 338.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 321.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 603.T + 9.12e5T^{2} \) |
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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
−9.383041991942432732782002040396, −8.544862007342993796262784682851, −7.57305399255513178957111028452, −6.93858906184399569670443519369, −5.98036539433953541771008329532, −4.50980435078596149153393392321, −3.99579980259020672708600674621, −3.09185549636127751608683289376, −1.80455324632707966211183109054, 0,
1.80455324632707966211183109054, 3.09185549636127751608683289376, 3.99579980259020672708600674621, 4.50980435078596149153393392321, 5.98036539433953541771008329532, 6.93858906184399569670443519369, 7.57305399255513178957111028452, 8.544862007342993796262784682851, 9.383041991942432732782002040396