L(s) = 1 | − 3-s + 2·7-s + 9-s − 11-s + 4·13-s − 4·17-s − 2·21-s + 4·23-s − 27-s − 6·29-s − 8·31-s + 33-s + 2·37-s − 4·39-s − 10·41-s − 10·43-s + 12·47-s − 3·49-s + 4·51-s − 6·53-s − 12·59-s + 10·61-s + 2·63-s + 8·67-s − 4·69-s − 4·73-s − 2·77-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.970·17-s − 0.436·21-s + 0.834·23-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.174·33-s + 0.328·37-s − 0.640·39-s − 1.56·41-s − 1.52·43-s + 1.75·47-s − 3/7·49-s + 0.560·51-s − 0.824·53-s − 1.56·59-s + 1.28·61-s + 0.251·63-s + 0.977·67-s − 0.481·69-s − 0.468·73-s − 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.58138882615537302077912866322, −6.91629108327905975578105718929, −6.23730722537473010462921084945, −5.41640289599431210707225439731, −4.93742356452597272954736074943, −4.05405518472360767093942469164, −3.32000261701388629858950816448, −2.06796228809343510908551525820, −1.34581988556484141737853608543, 0,
1.34581988556484141737853608543, 2.06796228809343510908551525820, 3.32000261701388629858950816448, 4.05405518472360767093942469164, 4.93742356452597272954736074943, 5.41640289599431210707225439731, 6.23730722537473010462921084945, 6.91629108327905975578105718929, 7.58138882615537302077912866322