L(s) = 1 | − 5-s + 2·7-s − 3·9-s − 11-s − 4·13-s − 4·17-s + 25-s − 6·29-s − 2·35-s − 2·37-s + 6·41-s − 2·43-s + 3·45-s − 3·49-s − 10·53-s + 55-s − 12·59-s − 6·61-s − 6·63-s + 4·65-s + 12·67-s − 16·71-s + 4·73-s − 2·77-s + 4·79-s + 9·81-s − 2·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.755·7-s − 9-s − 0.301·11-s − 1.10·13-s − 0.970·17-s + 1/5·25-s − 1.11·29-s − 0.338·35-s − 0.328·37-s + 0.937·41-s − 0.304·43-s + 0.447·45-s − 3/7·49-s − 1.37·53-s + 0.134·55-s − 1.56·59-s − 0.768·61-s − 0.755·63-s + 0.496·65-s + 1.46·67-s − 1.89·71-s + 0.468·73-s − 0.227·77-s + 0.450·79-s + 81-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.612910329107483789668876252462, −8.834090043318512684287621245587, −7.978734464208958320503765844199, −7.36247134612527590927307037370, −6.21087114261569312686272580596, −5.17000472187588479255187728599, −4.45821261643777188910209572887, −3.12595338350487188300141328594, −2.03399759604601396643465282622, 0,
2.03399759604601396643465282622, 3.12595338350487188300141328594, 4.45821261643777188910209572887, 5.17000472187588479255187728599, 6.21087114261569312686272580596, 7.36247134612527590927307037370, 7.978734464208958320503765844199, 8.834090043318512684287621245587, 9.612910329107483789668876252462