| L(s) = 1 | − 15.2·3-s + 43.0·5-s + 22.6·7-s − 10.9·9-s + 258.·11-s − 990.·13-s − 656.·15-s − 218·17-s + 1.99e3·19-s − 344.·21-s + 3.41e3·23-s − 1.26e3·25-s + 3.86e3·27-s + 4.86e3·29-s − 8.68e3·31-s − 3.94e3·33-s + 974.·35-s − 3.23e3·37-s + 1.50e4·39-s − 8.41e3·41-s + 685.·43-s − 473.·45-s − 1.81e4·47-s − 1.62e4·49-s + 3.32e3·51-s + 2.65e4·53-s + 1.11e4·55-s + ⋯ |
| L(s) = 1 | − 0.977·3-s + 0.770·5-s + 0.174·7-s − 0.0452·9-s + 0.645·11-s − 1.62·13-s − 0.753·15-s − 0.182·17-s + 1.26·19-s − 0.170·21-s + 1.34·23-s − 0.406·25-s + 1.02·27-s + 1.07·29-s − 1.62·31-s − 0.630·33-s + 0.134·35-s − 0.388·37-s + 1.58·39-s − 0.781·41-s + 0.0565·43-s − 0.0348·45-s − 1.19·47-s − 0.969·49-s + 0.178·51-s + 1.29·53-s + 0.497·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 15.2T + 243T^{2} \) |
| 5 | \( 1 - 43.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 22.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 258.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 990.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 218T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.99e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.23e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 685.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.81e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.65e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.37e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.73e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.06e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.38e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.17e4T + 8.58e9T^{2} \) |
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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
−10.73628038338790674405810283839, −9.781531301639600594583657902059, −8.997981937093363826474457854776, −7.45806101086485829743557141857, −6.55495604008638997225316767466, −5.44973410353093154609656472453, −4.82089962414029444262605402629, −2.97122999768622557308585713520, −1.47476286116198086991014712465, 0,
1.47476286116198086991014712465, 2.97122999768622557308585713520, 4.82089962414029444262605402629, 5.44973410353093154609656472453, 6.55495604008638997225316767466, 7.45806101086485829743557141857, 8.997981937093363826474457854776, 9.781531301639600594583657902059, 10.73628038338790674405810283839
