L(s) = 1 | − 2·4-s + 5-s − 2·7-s + 4·11-s − 3·13-s + 4·16-s − 4·19-s − 2·20-s + 3·23-s − 4·25-s + 4·28-s − 5·29-s − 5·31-s − 2·35-s + 5·37-s − 4·41-s − 4·43-s − 8·44-s − 12·47-s − 3·49-s + 6·52-s + 53-s + 4·55-s − 5·59-s − 5·61-s − 8·64-s − 3·65-s + ⋯ |
L(s) = 1 | − 4-s + 0.447·5-s − 0.755·7-s + 1.20·11-s − 0.832·13-s + 16-s − 0.917·19-s − 0.447·20-s + 0.625·23-s − 4/5·25-s + 0.755·28-s − 0.928·29-s − 0.898·31-s − 0.338·35-s + 0.821·37-s − 0.624·41-s − 0.609·43-s − 1.20·44-s − 1.75·47-s − 3/7·49-s + 0.832·52-s + 0.137·53-s + 0.539·55-s − 0.650·59-s − 0.640·61-s − 64-s − 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 127 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−9.523113827009498028071402658877, −8.825856503308691459233915162207, −7.83198544949496502505532924958, −6.77033864632910218836064236842, −6.03192316257116187625035852646, −5.04638773734818414532746887552, −4.11601093618984510481782697790, −3.24715927499517921073847449407, −1.72536667495706067705337718798, 0,
1.72536667495706067705337718798, 3.24715927499517921073847449407, 4.11601093618984510481782697790, 5.04638773734818414532746887552, 6.03192316257116187625035852646, 6.77033864632910218836064236842, 7.83198544949496502505532924958, 8.825856503308691459233915162207, 9.523113827009498028071402658877