L(s) = 1 | + 5.31·2-s − 3.71·4-s + 103.·5-s − 189.·8-s + 550.·10-s − 653.·11-s + 138.·13-s − 891.·16-s + 1.17e3·17-s + 1.71e3·19-s − 384.·20-s − 3.47e3·22-s + 4.02e3·23-s + 7.58e3·25-s + 734.·26-s − 2.64e3·29-s + 2.87e3·31-s + 1.33e3·32-s + 6.24e3·34-s + 2.85e3·37-s + 9.09e3·38-s − 1.96e4·40-s + 216.·41-s + 2.92e3·43-s + 2.42e3·44-s + 2.13e4·46-s + 1.48e4·47-s + ⋯ |
L(s) = 1 | + 0.940·2-s − 0.116·4-s + 1.85·5-s − 1.04·8-s + 1.74·10-s − 1.62·11-s + 0.226·13-s − 0.870·16-s + 0.985·17-s + 1.08·19-s − 0.215·20-s − 1.53·22-s + 1.58·23-s + 2.42·25-s + 0.212·26-s − 0.585·29-s + 0.537·31-s + 0.231·32-s + 0.926·34-s + 0.343·37-s + 1.02·38-s − 1.94·40-s + 0.0201·41-s + 0.241·43-s + 0.189·44-s + 1.48·46-s + 0.978·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(4.294691190\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.294691190\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 5.31T + 32T^{2} \) |
| 5 | \( 1 - 103.T + 3.12e3T^{2} \) |
| 11 | \( 1 + 653.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 138.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.17e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.71e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 216.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.48e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.75e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.55e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.62e4T + 8.58e9T^{2} \) |
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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
−10.17407961701712621996829745107, −9.586998262705515428248263924528, −8.686430263543575563575281851407, −7.36281059449699883745627711740, −6.09055580703777371985073619288, −5.40172447670850959456389475806, −4.95709707252150750463409143253, −3.21023858620297044783819171690, −2.47464217759359565939818430451, −0.967476634898867691799064602154,
0.967476634898867691799064602154, 2.47464217759359565939818430451, 3.21023858620297044783819171690, 4.95709707252150750463409143253, 5.40172447670850959456389475806, 6.09055580703777371985073619288, 7.36281059449699883745627711740, 8.686430263543575563575281851407, 9.586998262705515428248263924528, 10.17407961701712621996829745107