| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 7·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s − 14·10-s − 21·11-s + 12·12-s − 9·13-s + 14·14-s + 21·15-s + 16·16-s − 17·17-s − 18·18-s + 19-s + 28·20-s − 21·21-s + 42·22-s + 51·23-s − 24·24-s − 76·25-s + 18·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.626·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.442·10-s − 0.575·11-s + 0.288·12-s − 0.192·13-s + 0.267·14-s + 0.361·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.0120·19-s + 0.313·20-s − 0.218·21-s + 0.407·22-s + 0.462·23-s − 0.204·24-s − 0.607·25-s + 0.135·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| 17 | \( 1 + p T \) |
| good | 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 21 T + p^{3} T^{2} \) |
| 13 | \( 1 + 9 T + p^{3} T^{2} \) |
| 19 | \( 1 - T + p^{3} T^{2} \) |
| 23 | \( 1 - 51 T + p^{3} T^{2} \) |
| 29 | \( 1 + 266 T + p^{3} T^{2} \) |
| 31 | \( 1 + 30 T + p^{3} T^{2} \) |
| 37 | \( 1 + 376 T + p^{3} T^{2} \) |
| 41 | \( 1 - 85 T + p^{3} T^{2} \) |
| 43 | \( 1 - 227 T + p^{3} T^{2} \) |
| 47 | \( 1 + 10 T + p^{3} T^{2} \) |
| 53 | \( 1 - 122 T + p^{3} T^{2} \) |
| 59 | \( 1 + 666 T + p^{3} T^{2} \) |
| 61 | \( 1 + 184 T + p^{3} T^{2} \) |
| 67 | \( 1 + 996 T + p^{3} T^{2} \) |
| 71 | \( 1 - 972 T + p^{3} T^{2} \) |
| 73 | \( 1 - 398 T + p^{3} T^{2} \) |
| 79 | \( 1 - 190 T + p^{3} T^{2} \) |
| 83 | \( 1 + 894 T + p^{3} T^{2} \) |
| 89 | \( 1 - 596 T + p^{3} T^{2} \) |
| 97 | \( 1 - 756 T + p^{3} T^{2} \) |
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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
−9.402173302201650847892673068570, −9.001378353860813260271604030288, −7.86924906299191490216514174216, −7.20704032239762420591538489834, −6.15547787169496061489477624933, −5.20233118465514437135557641872, −3.72357299747270974141120062695, −2.60276003899022724949638765224, −1.65897159624120154977263452776, 0,
1.65897159624120154977263452776, 2.60276003899022724949638765224, 3.72357299747270974141120062695, 5.20233118465514437135557641872, 6.15547787169496061489477624933, 7.20704032239762420591538489834, 7.86924906299191490216514174216, 9.001378353860813260271604030288, 9.402173302201650847892673068570
