| L(s) = 1 | − 21.5·2-s + 28.3·3-s + 338.·4-s − 473.·5-s − 612.·6-s + 99.1·7-s − 4.54e3·8-s − 1.38e3·9-s + 1.02e4·10-s + 3.43e3·11-s + 9.59e3·12-s − 2.14e3·14-s − 1.34e4·15-s + 5.47e4·16-s + 2.28e4·17-s + 2.98e4·18-s + 3.69e3·19-s − 1.60e5·20-s + 2.81e3·21-s − 7.41e4·22-s − 6.89e4·23-s − 1.28e5·24-s + 1.46e5·25-s − 1.01e5·27-s + 3.35e4·28-s − 1.42e4·29-s + 2.90e5·30-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 0.606·3-s + 2.64·4-s − 1.69·5-s − 1.15·6-s + 0.109·7-s − 3.13·8-s − 0.631·9-s + 3.23·10-s + 0.777·11-s + 1.60·12-s − 0.208·14-s − 1.02·15-s + 3.34·16-s + 1.13·17-s + 1.20·18-s + 0.123·19-s − 4.47·20-s + 0.0663·21-s − 1.48·22-s − 1.18·23-s − 1.90·24-s + 1.87·25-s − 0.990·27-s + 0.288·28-s − 0.108·29-s + 1.96·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| good | 2 | \( 1 + 21.5T + 128T^{2} \) |
| 3 | \( 1 - 28.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 473.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 99.1T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.43e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.28e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.69e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.89e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.42e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.63e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.17e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.64e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.13e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.87e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.48e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.73e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.80e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.14e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.21e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.79e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 9.39e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.55e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.34e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.92e6T + 8.07e13T^{2} \) |
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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
−10.91063470224244438098741322025, −9.660019875812038829040561426008, −8.792469215521165731948368212463, −7.88439249714487204854987188955, −7.61507757257929170835361911316, −6.17505930660187662406110104419, −3.82040934918995136057647932000, −2.70900190793780142010406581558, −1.09807136588851126545503552799, 0,
1.09807136588851126545503552799, 2.70900190793780142010406581558, 3.82040934918995136057647932000, 6.17505930660187662406110104419, 7.61507757257929170835361911316, 7.88439249714487204854987188955, 8.792469215521165731948368212463, 9.660019875812038829040561426008, 10.91063470224244438098741322025
