| L(s) = 1 | − 4·5-s + 7-s + 4·11-s − 13-s − 3·17-s − 2·19-s − 3·23-s + 11·25-s + 9·29-s + 31-s − 4·35-s + 10·37-s − 10·41-s + 43-s − 10·47-s + 49-s − 9·53-s − 16·55-s − 3·59-s − 6·61-s + 4·65-s + 13·67-s − 13·71-s + 10·73-s + 4·77-s − 10·79-s + 4·83-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.377·7-s + 1.20·11-s − 0.277·13-s − 0.727·17-s − 0.458·19-s − 0.625·23-s + 11/5·25-s + 1.67·29-s + 0.179·31-s − 0.676·35-s + 1.64·37-s − 1.56·41-s + 0.152·43-s − 1.45·47-s + 1/7·49-s − 1.23·53-s − 2.15·55-s − 0.390·59-s − 0.768·61-s + 0.496·65-s + 1.58·67-s − 1.54·71-s + 1.17·73-s + 0.455·77-s − 1.12·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.185517151153628976930591011364, −7.83842775038506891881006809464, −6.73822641662859995260570903588, −6.43889285444342074692739085950, −4.90674959884917050452138575272, −4.36893933283345036510361285272, −3.74232463586956221426890519693, −2.75355482851484232495944630799, −1.32153555162149049648272418092, 0,
1.32153555162149049648272418092, 2.75355482851484232495944630799, 3.74232463586956221426890519693, 4.36893933283345036510361285272, 4.90674959884917050452138575272, 6.43889285444342074692739085950, 6.73822641662859995260570903588, 7.83842775038506891881006809464, 8.185517151153628976930591011364
