| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 7.13·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s − 14.2·10-s − 7.09·11-s − 12·12-s + 0.279·13-s + 14·14-s − 21.4·15-s + 16·16-s + 27.8·17-s − 18·18-s + 19·19-s + 28.5·20-s + 21·21-s + 14.1·22-s + 6.59·23-s + 24·24-s − 74.0·25-s − 0.559·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.638·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.451·10-s − 0.194·11-s − 0.288·12-s + 0.00596·13-s + 0.267·14-s − 0.368·15-s + 0.250·16-s + 0.397·17-s − 0.235·18-s + 0.229·19-s + 0.319·20-s + 0.218·21-s + 0.137·22-s + 0.0597·23-s + 0.204·24-s − 0.592·25-s − 0.00422·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 19 | \( 1 - 19T \) |
| good | 5 | \( 1 - 7.13T + 125T^{2} \) |
| 11 | \( 1 + 7.09T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.279T + 2.19e3T^{2} \) |
| 17 | \( 1 - 27.8T + 4.91e3T^{2} \) |
| 23 | \( 1 - 6.59T + 1.21e4T^{2} \) |
| 29 | \( 1 + 277.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 326.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 129.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 296.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 148.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 455.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 459.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 145.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 961.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 889.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 104.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 745.T + 9.12e5T^{2} \) |
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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
−9.782876424371429514642898482996, −8.709678560462165098523147038106, −7.71067162075241610914365519508, −6.86381376076701580941876726947, −5.96228503377750605546962865940, −5.29224836207262156521762534106, −3.85104824517842829656910576259, −2.52075894700835529630104633740, −1.32915809657444210904725241169, 0,
1.32915809657444210904725241169, 2.52075894700835529630104633740, 3.85104824517842829656910576259, 5.29224836207262156521762534106, 5.96228503377750605546962865940, 6.86381376076701580941876726947, 7.71067162075241610914365519508, 8.709678560462165098523147038106, 9.782876424371429514642898482996
