| L(s) = 1 | + 3-s − 2·4-s − 7-s + 9-s − 11-s − 2·12-s − 13-s + 4·16-s − 6·17-s − 7·19-s − 21-s + 6·23-s + 27-s + 2·28-s − 6·29-s − 7·31-s − 33-s − 2·36-s + 2·37-s − 39-s − 6·41-s − 43-s + 2·44-s + 4·48-s − 6·49-s − 6·51-s + 2·52-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 0.277·13-s + 16-s − 1.45·17-s − 1.60·19-s − 0.218·21-s + 1.25·23-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 1.25·31-s − 0.174·33-s − 1/3·36-s + 0.328·37-s − 0.160·39-s − 0.937·41-s − 0.152·43-s + 0.301·44-s + 0.577·48-s − 6/7·49-s − 0.840·51-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−9.567309717603522064679693310635, −8.955732103392000591141257065944, −8.367781725670412632475845953636, −7.30708842123598942599324121736, −6.37594318082116078700438020966, −5.13071370263278861735871300481, −4.31466281424388077222299797352, −3.35971135428784856363070844882, −2.04859260711079521330868103173, 0,
2.04859260711079521330868103173, 3.35971135428784856363070844882, 4.31466281424388077222299797352, 5.13071370263278861735871300481, 6.37594318082116078700438020966, 7.30708842123598942599324121736, 8.367781725670412632475845953636, 8.955732103392000591141257065944, 9.567309717603522064679693310635
