| L(s) = 1 | − 9·3-s + 94.1·5-s − 33.4·7-s + 81·9-s − 109.·11-s − 169·13-s − 847.·15-s + 1.15e3·17-s − 1.31e3·19-s + 301.·21-s − 52.2·23-s + 5.73e3·25-s − 729·27-s − 7.09e3·29-s − 8.21e3·31-s + 988.·33-s − 3.14e3·35-s − 1.28e4·37-s + 1.52e3·39-s + 2.81e3·41-s − 3.49e3·43-s + 7.62e3·45-s + 4.23e3·47-s − 1.56e4·49-s − 1.03e4·51-s + 1.35e4·53-s − 1.03e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.68·5-s − 0.258·7-s + 0.333·9-s − 0.273·11-s − 0.277·13-s − 0.972·15-s + 0.969·17-s − 0.834·19-s + 0.149·21-s − 0.0205·23-s + 1.83·25-s − 0.192·27-s − 1.56·29-s − 1.53·31-s + 0.157·33-s − 0.434·35-s − 1.53·37-s + 0.160·39-s + 0.261·41-s − 0.288·43-s + 0.561·45-s + 0.279·47-s − 0.933·49-s − 0.559·51-s + 0.661·53-s − 0.460·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 13 | \( 1 + 169T \) |
| good | 5 | \( 1 - 94.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 33.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + 109.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.15e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 52.2T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.81e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.49e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.23e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.26e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.00e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.82e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.12e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.25e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.86e3T + 8.58e9T^{2} \) |
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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
−9.578908631620483307790933046459, −8.772635105826026595770429892442, −7.41207973736681211725766963521, −6.54808797305051215125570994080, −5.60378161056747434002530798638, −5.24320583567421173657447011017, −3.69738870379290766701756430400, −2.31381533049260790471630204992, −1.47443454240362464698813516608, 0,
1.47443454240362464698813516608, 2.31381533049260790471630204992, 3.69738870379290766701756430400, 5.24320583567421173657447011017, 5.60378161056747434002530798638, 6.54808797305051215125570994080, 7.41207973736681211725766963521, 8.772635105826026595770429892442, 9.578908631620483307790933046459
