| L(s) = 1 | − 12.7·2-s − 48.2·3-s + 34.2·4-s − 506.·5-s + 615.·6-s − 343·7-s + 1.19e3·8-s + 145.·9-s + 6.45e3·10-s + 1.35e3·11-s − 1.65e3·12-s + 2.19e3·13-s + 4.36e3·14-s + 2.44e4·15-s − 1.95e4·16-s + 1.05e4·17-s − 1.85e3·18-s + 1.42e4·19-s − 1.73e4·20-s + 1.65e4·21-s − 1.72e4·22-s − 8.97e3·23-s − 5.76e4·24-s + 1.78e5·25-s − 2.79e4·26-s + 9.85e4·27-s − 1.17e4·28-s + ⋯ |
| L(s) = 1 | − 1.12·2-s − 1.03·3-s + 0.267·4-s − 1.81·5-s + 1.16·6-s − 0.377·7-s + 0.824·8-s + 0.0664·9-s + 2.04·10-s + 0.306·11-s − 0.276·12-s + 0.277·13-s + 0.425·14-s + 1.87·15-s − 1.19·16-s + 0.520·17-s − 0.0748·18-s + 0.475·19-s − 0.485·20-s + 0.390·21-s − 0.344·22-s − 0.153·23-s − 0.851·24-s + 2.28·25-s − 0.312·26-s + 0.964·27-s − 0.101·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 2 | \( 1 + 12.7T + 128T^{2} \) |
| 3 | \( 1 + 48.2T + 2.18e3T^{2} \) |
| 5 | \( 1 + 506.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 1.35e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.05e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.42e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.97e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.38e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.09e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.49e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.28e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.73e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.24e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.40e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.73e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.75e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.37e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.17e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.53e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.35e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.79e4T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.49e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.51e7T + 8.07e13T^{2} \) |
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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
−11.67820268245597931510164622316, −11.17855517029504857121654087873, −10.01938936468146411730338518755, −8.651384755340530243315931916254, −7.77529226520526045069556873353, −6.68780842844192095448503139421, −4.94192139842278689718592259196, −3.60085473309128521613366773934, −0.888522798364425778889058611502, 0,
0.888522798364425778889058611502, 3.60085473309128521613366773934, 4.94192139842278689718592259196, 6.68780842844192095448503139421, 7.77529226520526045069556873353, 8.651384755340530243315931916254, 10.01938936468146411730338518755, 11.17855517029504857121654087873, 11.67820268245597931510164622316
