| L(s) = 1 | + 16.3·2-s + 83.4·3-s + 140.·4-s − 338.·5-s + 1.36e3·6-s + 736.·7-s + 208.·8-s + 4.78e3·9-s − 5.55e3·10-s − 3.21e3·11-s + 1.17e4·12-s − 9.07e3·13-s + 1.20e4·14-s − 2.82e4·15-s − 1.45e4·16-s − 2.16e4·17-s + 7.84e4·18-s − 4.14e4·19-s − 4.76e4·20-s + 6.15e4·21-s − 5.26e4·22-s + 1.74e4·24-s + 3.65e4·25-s − 1.48e5·26-s + 2.16e5·27-s + 1.03e5·28-s + 1.86e5·29-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 1.78·3-s + 1.09·4-s − 1.21·5-s + 2.58·6-s + 0.811·7-s + 0.144·8-s + 2.18·9-s − 1.75·10-s − 0.727·11-s + 1.96·12-s − 1.14·13-s + 1.17·14-s − 2.16·15-s − 0.890·16-s − 1.06·17-s + 3.16·18-s − 1.38·19-s − 1.33·20-s + 1.44·21-s − 1.05·22-s + 0.257·24-s + 0.467·25-s − 1.65·26-s + 2.11·27-s + 0.892·28-s + 1.41·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 - 16.3T + 128T^{2} \) |
| 3 | \( 1 - 83.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + 338.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 736.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 9.07e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.16e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.14e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.86e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.69e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.35e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.82e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.35e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.13e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.20e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.87e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 258.T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.36e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.80e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.70e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.49e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.75e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.50e6T + 8.07e13T^{2} \) |
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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
−8.865642111247702450156586692893, −8.316644227582075987879550587241, −7.51450103490722625070790214587, −6.70808535727744241252395415070, −4.90366036283727011341777497756, −4.43039664888676795737810599982, −3.63772717655920161395742550324, −2.65533665013682409831506103351, −2.04813754427201995164438517527, 0,
2.04813754427201995164438517527, 2.65533665013682409831506103351, 3.63772717655920161395742550324, 4.43039664888676795737810599982, 4.90366036283727011341777497756, 6.70808535727744241252395415070, 7.51450103490722625070790214587, 8.316644227582075987879550587241, 8.865642111247702450156586692893
