| L(s) = 1 | + 2-s − 3-s − 4-s − 5-s − 6-s + 2·7-s − 3·8-s + 9-s − 10-s − 2·11-s + 12-s + 2·14-s + 15-s − 16-s − 7·17-s + 18-s − 6·19-s + 20-s − 2·21-s − 2·22-s − 6·23-s + 3·24-s − 4·25-s − 27-s − 2·28-s − 29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.534·14-s + 0.258·15-s − 1/4·16-s − 1.69·17-s + 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.436·21-s − 0.426·22-s − 1.25·23-s + 0.612·24-s − 4/5·25-s − 0.192·27-s − 0.377·28-s − 0.185·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.79076199904715533558890862998, −9.621109724690854729571825572923, −8.551927544281445958636300086280, −7.83924221365301722237117171435, −6.47571815039088450225769754472, −5.65964626382136303973235650945, −4.47621499216397670225811127054, −4.16029256597568730693149405009, −2.32792075963527314368769910890, 0,
2.32792075963527314368769910890, 4.16029256597568730693149405009, 4.47621499216397670225811127054, 5.65964626382136303973235650945, 6.47571815039088450225769754472, 7.83924221365301722237117171435, 8.551927544281445958636300086280, 9.621109724690854729571825572923, 10.79076199904715533558890862998
