| L(s) = 1 | − 12.4·2-s + 27.4·4-s + 40.8·5-s + 1.12e3·7-s + 1.25e3·8-s − 509.·10-s − 500.·11-s − 1.28e4·13-s − 1.40e4·14-s − 1.91e4·16-s + 2.37e4·17-s − 3.28e4·19-s + 1.12e3·20-s + 6.23e3·22-s − 1.21e4·23-s − 7.64e4·25-s + 1.60e5·26-s + 3.09e4·28-s + 1.52e5·29-s + 2.52e5·31-s + 7.81e4·32-s − 2.96e5·34-s + 4.60e4·35-s + 8.94e4·37-s + 4.09e5·38-s + 5.12e4·40-s − 5.49e5·41-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.214·4-s + 0.146·5-s + 1.24·7-s + 0.865·8-s − 0.161·10-s − 0.113·11-s − 1.62·13-s − 1.36·14-s − 1.16·16-s + 1.17·17-s − 1.09·19-s + 0.0313·20-s + 0.124·22-s − 0.208·23-s − 0.978·25-s + 1.79·26-s + 0.266·28-s + 1.15·29-s + 1.52·31-s + 0.421·32-s − 1.29·34-s + 0.181·35-s + 0.290·37-s + 1.21·38-s + 0.126·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 + 12.4T + 128T^{2} \) |
| 5 | \( 1 - 40.8T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.12e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 500.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.28e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.37e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.28e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.52e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.52e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 8.94e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.49e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.30e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.34e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.73e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.58e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.40e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.57e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 7.09e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 7.97e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.98e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.20e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.23e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.54e6T + 8.07e13T^{2} \) |
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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
−10.16149676298347648264220949605, −9.896821511624559579855374263279, −8.389576104723308912570010184291, −8.023767723431977728838831350454, −6.91236919750194233849553592962, −5.23590192648070717499374856129, −4.39213204845544468683364100689, −2.38334305414622512033289138180, −1.30013875802404360395687104548, 0,
1.30013875802404360395687104548, 2.38334305414622512033289138180, 4.39213204845544468683364100689, 5.23590192648070717499374856129, 6.91236919750194233849553592962, 8.023767723431977728838831350454, 8.389576104723308912570010184291, 9.896821511624559579855374263279, 10.16149676298347648264220949605
