| L(s) = 1 | + 20.4·2-s − 14.6·3-s + 289.·4-s − 478.·5-s − 299.·6-s + 1.19e3·7-s + 3.30e3·8-s − 1.97e3·9-s − 9.78e3·10-s − 975.·11-s − 4.24e3·12-s + 2.44e4·14-s + 7.01e3·15-s + 3.05e4·16-s − 1.11e4·17-s − 4.03e4·18-s − 3.38e4·19-s − 1.38e5·20-s − 1.75e4·21-s − 1.99e4·22-s − 6.06e4·23-s − 4.84e4·24-s + 1.50e5·25-s + 6.09e4·27-s + 3.47e5·28-s − 1.75e5·29-s + 1.43e5·30-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 0.313·3-s + 2.26·4-s − 1.71·5-s − 0.566·6-s + 1.31·7-s + 2.28·8-s − 0.901·9-s − 3.09·10-s − 0.220·11-s − 0.709·12-s + 2.38·14-s + 0.536·15-s + 1.86·16-s − 0.550·17-s − 1.62·18-s − 1.13·19-s − 3.87·20-s − 0.413·21-s − 0.399·22-s − 1.03·23-s − 0.715·24-s + 1.93·25-s + 0.595·27-s + 2.98·28-s − 1.33·29-s + 0.969·30-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 | \( 1 - 20.4T + 128T^{2} \) |
| 3 | \( 1 + 14.6T + 2.18e3T^{2} \) |
| 5 | \( 1 + 478.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.19e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 975.T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.11e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.38e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.06e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.75e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.16e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.62e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.46e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.62e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.89e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.82e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.62e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 4.65e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.90e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.19e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.24e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.79e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.82e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.53e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.25e6T + 8.07e13T^{2} \) |
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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
−11.21853828526237418706911353755, −11.01088278961819674926924362828, −8.389063020276736791582170682494, −7.68539159225634252296198865816, −6.43261864422283993353271493435, −5.17920525004764851116922797338, −4.40722436154141741729724402615, −3.52871671395737188729736031175, −2.09453446316473483503500166482, 0,
2.09453446316473483503500166482, 3.52871671395737188729736031175, 4.40722436154141741729724402615, 5.17920525004764851116922797338, 6.43261864422283993353271493435, 7.68539159225634252296198865816, 8.389063020276736791582170682494, 11.01088278961819674926924362828, 11.21853828526237418706911353755
