L(s) = 1 | − 1.20e4·2-s − 4.78e6·3-s − 3.91e8·4-s − 3.60e9·5-s + 5.76e10·6-s + 6.78e11·7-s + 1.11e13·8-s + 2.28e13·9-s + 4.33e13·10-s + 2.47e15·11-s + 1.87e15·12-s + 1.86e16·13-s − 8.17e15·14-s + 1.72e16·15-s + 7.54e16·16-s − 1.63e17·17-s − 2.75e17·18-s + 4.78e18·19-s + 1.41e18·20-s − 3.24e18·21-s − 2.97e19·22-s + 8.72e19·23-s − 5.35e19·24-s − 1.73e20·25-s − 2.24e20·26-s − 1.09e20·27-s − 2.65e20·28-s + ⋯ |
L(s) = 1 | − 0.520·2-s − 0.577·3-s − 0.729·4-s − 0.263·5-s + 0.300·6-s + 0.377·7-s + 0.899·8-s + 0.333·9-s + 0.137·10-s + 1.96·11-s + 0.421·12-s + 1.31·13-s − 0.196·14-s + 0.152·15-s + 0.261·16-s − 0.235·17-s − 0.173·18-s + 1.37·19-s + 0.192·20-s − 0.218·21-s − 1.02·22-s + 1.57·23-s − 0.519·24-s − 0.930·25-s − 0.683·26-s − 0.192·27-s − 0.275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(15)\) |
\(\approx\) |
\(1.681689473\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.681689473\) |
\(L(\frac{31}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 4.78e6T \) |
| 7 | \( 1 - 6.78e11T \) |
good | 2 | \( 1 + 1.20e4T + 5.36e8T^{2} \) |
| 5 | \( 1 + 3.60e9T + 1.86e20T^{2} \) |
| 11 | \( 1 - 2.47e15T + 1.58e30T^{2} \) |
| 13 | \( 1 - 1.86e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + 1.63e17T + 4.81e35T^{2} \) |
| 19 | \( 1 - 4.78e18T + 1.21e37T^{2} \) |
| 23 | \( 1 - 8.72e19T + 3.09e39T^{2} \) |
| 29 | \( 1 - 4.12e20T + 2.56e42T^{2} \) |
| 31 | \( 1 - 2.72e21T + 1.77e43T^{2} \) |
| 37 | \( 1 - 8.47e21T + 3.00e45T^{2} \) |
| 41 | \( 1 - 1.03e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 9.09e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + 1.23e24T + 3.09e48T^{2} \) |
| 53 | \( 1 - 9.35e24T + 1.00e50T^{2} \) |
| 59 | \( 1 + 4.11e25T + 2.26e51T^{2} \) |
| 61 | \( 1 - 6.20e25T + 5.95e51T^{2} \) |
| 67 | \( 1 + 3.26e26T + 9.04e52T^{2} \) |
| 71 | \( 1 + 1.43e26T + 4.85e53T^{2} \) |
| 73 | \( 1 - 7.74e26T + 1.08e54T^{2} \) |
| 79 | \( 1 + 4.94e27T + 1.07e55T^{2} \) |
| 83 | \( 1 + 6.81e27T + 4.50e55T^{2} \) |
| 89 | \( 1 - 2.47e28T + 3.40e56T^{2} \) |
| 97 | \( 1 - 8.17e28T + 4.13e57T^{2} \) |
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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
−11.86981798797578897812008573897, −11.00303241296693674157700514204, −9.508092753641114212562853659208, −8.689242474880381561366509546368, −7.29300578712274018914198419664, −5.95141323979487359943065647334, −4.54634336422261425702059346578, −3.62925752638482997596555375395, −1.28960731849444808813443017682, −0.866817809499206927815907983602,
0.866817809499206927815907983602, 1.28960731849444808813443017682, 3.62925752638482997596555375395, 4.54634336422261425702059346578, 5.95141323979487359943065647334, 7.29300578712274018914198419664, 8.689242474880381561366509546368, 9.508092753641114212562853659208, 11.00303241296693674157700514204, 11.86981798797578897812008573897