| L(s) = 1 | − 2·2-s + 4·3-s + 4·4-s − 5·5-s − 8·6-s − 4·7-s − 8·8-s + 7·9-s + 10·10-s + 16·12-s + 8·14-s − 20·15-s + 16·16-s − 14·18-s − 20·20-s − 16·21-s + 44·23-s − 32·24-s + 25·25-s − 8·27-s − 16·28-s − 22·29-s + 40·30-s − 32·32-s + 20·35-s + 28·36-s + 40·40-s + ⋯ |
| L(s) = 1 | − 2-s + 4/3·3-s + 4-s − 5-s − 4/3·6-s − 4/7·7-s − 8-s + 7/9·9-s + 10-s + 4/3·12-s + 4/7·14-s − 4/3·15-s + 16-s − 7/9·18-s − 20-s − 0.761·21-s + 1.91·23-s − 4/3·24-s + 25-s − 0.296·27-s − 4/7·28-s − 0.758·29-s + 4/3·30-s − 32-s + 4/7·35-s + 7/9·36-s + 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7229650981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7229650981\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 3 | \( 1 - 4 T + p^{2} T^{2} \) |
| 7 | \( 1 + 4 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 - 44 T + p^{2} T^{2} \) |
| 29 | \( 1 + 22 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( 1 - 62 T + p^{2} T^{2} \) |
| 43 | \( 1 + 76 T + p^{2} T^{2} \) |
| 47 | \( 1 + 4 T + p^{2} T^{2} \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 58 T + p^{2} T^{2} \) |
| 67 | \( 1 - 116 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( 1 + 76 T + p^{2} T^{2} \) |
| 89 | \( 1 + 142 T + p^{2} T^{2} \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−18.63816406074365121508136781928, −16.80935138257150220250581254380, −15.57139549986060157799701779343, −14.72593067763324282271886119312, −12.85247684298092282880208510447, −11.18695936575511574955659648201, −9.454328650700016053216796555828, −8.381442062248796717095282373411, −7.15767411640600946713444214301, −3.18175657009207892084271250607,
3.18175657009207892084271250607, 7.15767411640600946713444214301, 8.381442062248796717095282373411, 9.454328650700016053216796555828, 11.18695936575511574955659648201, 12.85247684298092282880208510447, 14.72593067763324282271886119312, 15.57139549986060157799701779343, 16.80935138257150220250581254380, 18.63816406074365121508136781928
