L(s) = 1 | − 2.81·2-s + 3.33·3-s − 0.0697·4-s − 5·5-s − 9.39·6-s + 35.3·7-s + 22.7·8-s − 15.8·9-s + 14.0·10-s − 11·11-s − 0.232·12-s + 16.5·13-s − 99.6·14-s − 16.6·15-s − 63.4·16-s − 35.7·17-s + 44.6·18-s + 19·19-s + 0.348·20-s + 118.·21-s + 30.9·22-s − 96.3·23-s + 75.8·24-s + 25·25-s − 46.6·26-s − 143.·27-s − 2.46·28-s + ⋯ |
L(s) = 1 | − 0.995·2-s + 0.642·3-s − 0.00871·4-s − 0.447·5-s − 0.639·6-s + 1.91·7-s + 1.00·8-s − 0.587·9-s + 0.445·10-s − 0.301·11-s − 0.00559·12-s + 0.353·13-s − 1.90·14-s − 0.287·15-s − 0.991·16-s − 0.509·17-s + 0.585·18-s + 0.229·19-s + 0.00389·20-s + 1.22·21-s + 0.300·22-s − 0.873·23-s + 0.644·24-s + 0.200·25-s − 0.351·26-s − 1.01·27-s − 0.0166·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
good | 2 | \( 1 + 2.81T + 8T^{2} \) |
| 3 | \( 1 - 3.33T + 27T^{2} \) |
| 7 | \( 1 - 35.3T + 343T^{2} \) |
| 13 | \( 1 - 16.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.7T + 4.91e3T^{2} \) |
| 23 | \( 1 + 96.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 163.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 130.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 7.26T + 5.06e4T^{2} \) |
| 41 | \( 1 + 315.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 250.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 280.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 504.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 49.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 360.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 370.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 42.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 771.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 363.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 0.664T + 7.04e5T^{2} \) |
| 97 | \( 1 + 573.T + 9.12e5T^{2} \) |
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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
−8.773868948064443065649660419730, −8.324536946128196805794703803554, −7.955259124614380181059579893441, −7.09942321473050177309614061847, −5.53193968267876566002908428307, −4.69094490856138721717637199674, −3.78456038516061115208535469764, −2.27244047346374304980837966031, −1.39690618399071779834530675370, 0,
1.39690618399071779834530675370, 2.27244047346374304980837966031, 3.78456038516061115208535469764, 4.69094490856138721717637199674, 5.53193968267876566002908428307, 7.09942321473050177309614061847, 7.955259124614380181059579893441, 8.324536946128196805794703803554, 8.773868948064443065649660419730