| L(s) = 1 | + 5-s + 2·7-s − 3·11-s − 4·13-s − 6·17-s − 7·19-s − 6·23-s + 25-s − 3·29-s + 5·31-s + 2·35-s − 4·37-s − 3·41-s + 8·43-s − 3·49-s + 6·53-s − 3·55-s + 3·59-s + 14·61-s − 4·65-s + 2·67-s − 15·71-s − 10·73-s − 6·77-s + 8·79-s − 6·85-s − 15·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.904·11-s − 1.10·13-s − 1.45·17-s − 1.60·19-s − 1.25·23-s + 1/5·25-s − 0.557·29-s + 0.898·31-s + 0.338·35-s − 0.657·37-s − 0.468·41-s + 1.21·43-s − 3/7·49-s + 0.824·53-s − 0.404·55-s + 0.390·59-s + 1.79·61-s − 0.496·65-s + 0.244·67-s − 1.78·71-s − 1.17·73-s − 0.683·77-s + 0.900·79-s − 0.650·85-s − 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.846133037232030187773342874921, −8.300317337572955138163027302389, −7.43135728541741986261852072880, −6.58557641701633406970309761410, −5.67914402549393928536297579517, −4.78912206678564536807129491168, −4.14988479247959547699462740057, −2.49699325668562889857088624426, −2.00575331118665702771246316335, 0,
2.00575331118665702771246316335, 2.49699325668562889857088624426, 4.14988479247959547699462740057, 4.78912206678564536807129491168, 5.67914402549393928536297579517, 6.58557641701633406970309761410, 7.43135728541741986261852072880, 8.300317337572955138163027302389, 8.846133037232030187773342874921
