| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 15.8·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 31.6·10-s + 22.2·11-s − 12·12-s + 48.9·13-s − 14·14-s + 47.5·15-s + 16·16-s + 85.3·17-s + 18·18-s − 19·19-s − 63.3·20-s + 21·21-s + 44.5·22-s − 148.·23-s − 24·24-s + 126.·25-s + 97.8·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.41·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.00·10-s + 0.610·11-s − 0.288·12-s + 1.04·13-s − 0.267·14-s + 0.818·15-s + 0.250·16-s + 1.21·17-s + 0.235·18-s − 0.229·19-s − 0.708·20-s + 0.218·21-s + 0.431·22-s − 1.34·23-s − 0.204·24-s + 1.00·25-s + 0.737·26-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 | \( 1 + 15.8T + 125T^{2} \) |
| 11 | \( 1 - 22.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 85.3T + 4.91e3T^{2} \) |
| 23 | \( 1 + 148.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 188.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 307.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 183.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 300.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 319.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 153.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 496.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 622.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 273.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 782.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.14e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 206.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 337.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.28T + 7.04e5T^{2} \) |
| 97 | \( 1 + 798.T + 9.12e5T^{2} \) |
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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
−9.587676473828122821854206668282, −8.316792299139728646824414884764, −7.68000199950861266777436106717, −6.63679827268039922293507577550, −5.97507467116774679703966365704, −4.82251852573238177752578366424, −3.83752181944026611338340830107, −3.35392259924847566571083575559, −1.42443890184877577340266162291, 0,
1.42443890184877577340266162291, 3.35392259924847566571083575559, 3.83752181944026611338340830107, 4.82251852573238177752578366424, 5.97507467116774679703966365704, 6.63679827268039922293507577550, 7.68000199950861266777436106717, 8.316792299139728646824414884764, 9.587676473828122821854206668282
