| L(s) = 1 | + 4.84·2-s − 6.19·3-s + 15.4·4-s − 15.2·5-s − 29.9·6-s − 4.31·7-s + 35.9·8-s + 11.3·9-s − 73.7·10-s − 24.5·11-s − 95.5·12-s − 20.8·14-s + 94.4·15-s + 50.5·16-s − 127.·17-s + 55.1·18-s − 51.7·19-s − 235.·20-s + 26.7·21-s − 118.·22-s + 87.3·23-s − 222.·24-s + 107.·25-s + 96.6·27-s − 66.5·28-s + 225.·29-s + 457.·30-s + ⋯ |
| L(s) = 1 | + 1.71·2-s − 1.19·3-s + 1.92·4-s − 1.36·5-s − 2.04·6-s − 0.233·7-s + 1.58·8-s + 0.422·9-s − 2.33·10-s − 0.673·11-s − 2.29·12-s − 0.398·14-s + 1.62·15-s + 0.790·16-s − 1.81·17-s + 0.722·18-s − 0.624·19-s − 2.62·20-s + 0.277·21-s − 1.15·22-s + 0.792·23-s − 1.89·24-s + 0.858·25-s + 0.689·27-s − 0.449·28-s + 1.44·29-s + 2.78·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| good | 2 | \( 1 - 4.84T + 8T^{2} \) |
| 3 | \( 1 + 6.19T + 27T^{2} \) |
| 5 | \( 1 + 15.2T + 125T^{2} \) |
| 7 | \( 1 + 4.31T + 343T^{2} \) |
| 11 | \( 1 + 24.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 127.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 51.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 87.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 225.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 108.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 115.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 191.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 123.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 36.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 119.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 804.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 678.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 87.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 981.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 263.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 321.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 344.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 482.T + 9.12e5T^{2} \) |
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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
−11.89784893130351919104784461859, −11.29225001127052302536953066938, −10.61128524962644706026558742802, −8.436412185901787450958582674099, −6.96584226197894634116259381501, −6.27888630699402961186922398671, −4.92734060149767745063162582877, −4.34032769665221031785013500075, −2.88312659173976791387873641473, 0,
2.88312659173976791387873641473, 4.34032769665221031785013500075, 4.92734060149767745063162582877, 6.27888630699402961186922398671, 6.96584226197894634116259381501, 8.436412185901787450958582674099, 10.61128524962644706026558742802, 11.29225001127052302536953066938, 11.89784893130351919104784461859
