| L(s) = 1 | − 3.52·2-s + 4.41·4-s + 5·5-s + 25.4·7-s + 12.6·8-s − 17.6·10-s − 71.3·11-s − 51.3·13-s − 89.6·14-s − 79.8·16-s − 33.3·17-s + 113.·19-s + 22.0·20-s + 251.·22-s + 81.9·23-s + 25·25-s + 181.·26-s + 112.·28-s − 246.·29-s + 222.·31-s + 180.·32-s + 117.·34-s + 127.·35-s + 22.3·37-s − 399.·38-s + 63.1·40-s − 434.·41-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.551·4-s + 0.447·5-s + 1.37·7-s + 0.558·8-s − 0.557·10-s − 1.95·11-s − 1.09·13-s − 1.71·14-s − 1.24·16-s − 0.475·17-s + 1.36·19-s + 0.246·20-s + 2.43·22-s + 0.743·23-s + 0.200·25-s + 1.36·26-s + 0.757·28-s − 1.58·29-s + 1.29·31-s + 0.995·32-s + 0.592·34-s + 0.614·35-s + 0.0994·37-s − 1.70·38-s + 0.249·40-s − 1.65·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 + 3.52T + 8T^{2} \) |
| 7 | \( 1 - 25.4T + 343T^{2} \) |
| 11 | \( 1 + 71.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 51.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 33.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 81.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 246.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 22.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 236.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 107.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 171.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 79.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 611.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 511.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 410.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 793.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 270.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 177.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 881.T + 9.12e5T^{2} \) |
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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
−10.22470468631406134068744240364, −9.551803346719183138579553726493, −8.444924049108022118121680814566, −7.79613708319798334569597549043, −7.14811395945593464516058885345, −5.25270629881691821845581184062, −4.86228557131038574859440496527, −2.64268932029356649570775336587, −1.54718867321210697574949638969, 0,
1.54718867321210697574949638969, 2.64268932029356649570775336587, 4.86228557131038574859440496527, 5.25270629881691821845581184062, 7.14811395945593464516058885345, 7.79613708319798334569597549043, 8.444924049108022118121680814566, 9.551803346719183138579553726493, 10.22470468631406134068744240364
