| L(s) = 1 | − 5-s + 2·7-s + 2·11-s − 4·13-s + 2·17-s − 4·19-s − 8·23-s + 25-s − 10·29-s + 4·31-s − 2·35-s + 8·43-s − 8·47-s − 3·49-s + 6·53-s − 2·55-s − 14·59-s + 14·61-s + 4·65-s + 4·67-s − 12·71-s + 6·73-s + 4·77-s − 12·79-s + 4·83-s − 2·85-s + 12·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.603·11-s − 1.10·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 1.85·29-s + 0.718·31-s − 0.338·35-s + 1.21·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s − 0.269·55-s − 1.82·59-s + 1.79·61-s + 0.496·65-s + 0.488·67-s − 1.42·71-s + 0.702·73-s + 0.455·77-s − 1.35·79-s + 0.439·83-s − 0.216·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.177735510636912404093715069618, −7.79979523582491151568952175531, −6.98865086758663648112324447119, −6.08823170975999641178845307383, −5.25494085780447371206947501442, −4.35719700865609900059773798869, −3.79057490301116406348088551581, −2.49538158301161918022879918142, −1.58792070167940625124508980254, 0,
1.58792070167940625124508980254, 2.49538158301161918022879918142, 3.79057490301116406348088551581, 4.35719700865609900059773798869, 5.25494085780447371206947501442, 6.08823170975999641178845307383, 6.98865086758663648112324447119, 7.79979523582491151568952175531, 8.177735510636912404093715069618
