L(s) = 1 | + 4.28·2-s + 2.85·3-s + 10.3·4-s − 6.36·5-s + 12.2·6-s + 29.4·7-s + 10.2·8-s − 18.8·9-s − 27.3·10-s + 61.3·11-s + 29.6·12-s + 18.5·13-s + 126.·14-s − 18.1·15-s − 39.1·16-s − 80.9·18-s + 115.·19-s − 66.1·20-s + 83.8·21-s + 263.·22-s − 7.38·23-s + 29.2·24-s − 84.4·25-s + 79.5·26-s − 130.·27-s + 305.·28-s + 164.·29-s + ⋯ |
L(s) = 1 | + 1.51·2-s + 0.548·3-s + 1.29·4-s − 0.569·5-s + 0.831·6-s + 1.58·7-s + 0.453·8-s − 0.699·9-s − 0.863·10-s + 1.68·11-s + 0.712·12-s + 0.395·13-s + 2.40·14-s − 0.312·15-s − 0.611·16-s − 1.05·18-s + 1.39·19-s − 0.739·20-s + 0.871·21-s + 2.55·22-s − 0.0669·23-s + 0.248·24-s − 0.675·25-s + 0.600·26-s − 0.932·27-s + 2.06·28-s + 1.05·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(5.274332694\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.274332694\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 - 4.28T + 8T^{2} \) |
| 3 | \( 1 - 2.85T + 27T^{2} \) |
| 5 | \( 1 + 6.36T + 125T^{2} \) |
| 7 | \( 1 - 29.4T + 343T^{2} \) |
| 11 | \( 1 - 61.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.5T + 2.19e3T^{2} \) |
| 19 | \( 1 - 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.38T + 1.21e4T^{2} \) |
| 29 | \( 1 - 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 127.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 158.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 31.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 157.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 460.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 166.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 343.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 112.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 984.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 524.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 852.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 201.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 22.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 502.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 680.T + 9.12e5T^{2} \) |
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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
−11.74532165424299281465313732868, −11.06928527277041658828144723293, −9.254824477567093811138476561539, −8.384662754078523613980733094494, −7.38750364228618963651939525510, −6.10907126308522856747023186701, −5.04729846575538108107802039703, −4.07901117388835591751870057524, −3.21825931649686990981130223866, −1.61944786017188430846660223186,
1.61944786017188430846660223186, 3.21825931649686990981130223866, 4.07901117388835591751870057524, 5.04729846575538108107802039703, 6.10907126308522856747023186701, 7.38750364228618963651939525510, 8.384662754078523613980733094494, 9.254824477567093811138476561539, 11.06928527277041658828144723293, 11.74532165424299281465313732868