| L(s) = 1 | − 5.52·2-s + 3·3-s + 22.4·4-s + 6.08·5-s − 16.5·6-s − 20.2·7-s − 80.0·8-s + 9·9-s − 33.5·10-s + 48.8·11-s + 67.4·12-s + 111.·14-s + 18.2·15-s + 262.·16-s − 37.7·17-s − 49.7·18-s − 120.·19-s + 136.·20-s − 60.8·21-s − 269.·22-s + 74.8·23-s − 240.·24-s − 88.0·25-s + 27·27-s − 456.·28-s − 112.·29-s − 100.·30-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 0.577·3-s + 2.81·4-s + 0.543·5-s − 1.12·6-s − 1.09·7-s − 3.53·8-s + 0.333·9-s − 1.06·10-s + 1.33·11-s + 1.62·12-s + 2.13·14-s + 0.314·15-s + 4.09·16-s − 0.538·17-s − 0.650·18-s − 1.45·19-s + 1.52·20-s − 0.631·21-s − 2.61·22-s + 0.678·23-s − 2.04·24-s − 0.704·25-s + 0.192·27-s − 3.07·28-s − 0.721·29-s − 0.613·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5.52T + 8T^{2} \) |
| 5 | \( 1 - 6.08T + 125T^{2} \) |
| 7 | \( 1 + 20.2T + 343T^{2} \) |
| 11 | \( 1 - 48.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 37.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 120.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 74.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 85.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 133.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 319.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 401.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 384.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 121.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 220.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 975.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 106.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 43.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 539.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 811.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.13e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 229.T + 9.12e5T^{2} \) |
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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
−9.716125657346578332865629553293, −9.167301524070757441263745039564, −8.681393894204845183816660822259, −7.48686275777391827635344128964, −6.61077475051217236407874608553, −6.11326066466291660911906801148, −3.72881508368956882778015543095, −2.49189862732150881736520245221, −1.51361866498344786989979103022, 0,
1.51361866498344786989979103022, 2.49189862732150881736520245221, 3.72881508368956882778015543095, 6.11326066466291660911906801148, 6.61077475051217236407874608553, 7.48686275777391827635344128964, 8.681393894204845183816660822259, 9.167301524070757441263745039564, 9.716125657346578332865629553293
