| L(s) = 1 | + 5-s − 7-s − 4·11-s − 2·17-s − 6·19-s − 6·23-s + 25-s + 2·31-s − 35-s + 2·37-s − 2·41-s + 4·43-s − 8·47-s + 49-s − 10·53-s − 4·55-s − 4·59-s − 2·61-s + 12·67-s − 8·71-s + 8·73-s + 4·77-s − 8·79-s + 4·83-s − 2·85-s − 10·89-s − 6·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.20·11-s − 0.485·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s + 0.359·31-s − 0.169·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 1.37·53-s − 0.539·55-s − 0.520·59-s − 0.256·61-s + 1.46·67-s − 0.949·71-s + 0.936·73-s + 0.455·77-s − 0.900·79-s + 0.439·83-s − 0.216·85-s − 1.05·89-s − 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.369191347224661880497031517925, −8.398677498267027029779046535759, −7.79228753450734108579484885860, −6.63026762565298849490087395711, −6.03837045133271924887513944710, −5.04152412655192654765370507928, −4.11686461055480055615061427630, −2.84984884496296869681057658548, −1.95387076547556235028224515009, 0,
1.95387076547556235028224515009, 2.84984884496296869681057658548, 4.11686461055480055615061427630, 5.04152412655192654765370507928, 6.03837045133271924887513944710, 6.63026762565298849490087395711, 7.79228753450734108579484885860, 8.398677498267027029779046535759, 9.369191347224661880497031517925
