L(s) = 1 | − 3·3-s − 3·5-s − 4·7-s + 6·9-s − 11-s + 3·13-s + 9·15-s + 2·17-s + 4·19-s + 12·21-s + 6·23-s + 4·25-s − 9·27-s + 29-s − 9·31-s + 3·33-s + 12·35-s + 8·37-s − 9·39-s − 8·41-s − 5·43-s − 18·45-s + 7·47-s + 9·49-s − 6·51-s + 5·53-s + 3·55-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 1.34·5-s − 1.51·7-s + 2·9-s − 0.301·11-s + 0.832·13-s + 2.32·15-s + 0.485·17-s + 0.917·19-s + 2.61·21-s + 1.25·23-s + 4/5·25-s − 1.73·27-s + 0.185·29-s − 1.61·31-s + 0.522·33-s + 2.02·35-s + 1.31·37-s − 1.44·39-s − 1.24·41-s − 0.762·43-s − 2.68·45-s + 1.02·47-s + 9/7·49-s − 0.840·51-s + 0.686·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.990402515185627546328567296310, −7.74730089520633784498196983232, −7.11856279275544935660492244768, −6.43560003277200267517245298001, −5.67479190582650462878626136824, −4.89577302719386106262640043369, −3.81759314249918795716653221706, −3.20774916762945711068213441442, −0.987975331709208295078797794960, 0,
0.987975331709208295078797794960, 3.20774916762945711068213441442, 3.81759314249918795716653221706, 4.89577302719386106262640043369, 5.67479190582650462878626136824, 6.43560003277200267517245298001, 7.11856279275544935660492244768, 7.74730089520633784498196983232, 8.990402515185627546328567296310