| L(s) = 1 | − 3-s + 5-s + 5·7-s − 26·9-s − 10·11-s + 13·13-s − 15-s + 93·17-s + 82·19-s − 5·21-s − 192·23-s − 124·25-s + 53·27-s + 106·29-s + 172·31-s + 10·33-s + 5·35-s − 379·37-s − 13·39-s − 148·41-s + 329·43-s − 26·45-s − 631·47-s − 318·49-s − 93·51-s − 160·53-s − 10·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s + 0.0894·5-s + 0.269·7-s − 0.962·9-s − 0.274·11-s + 0.277·13-s − 0.0172·15-s + 1.32·17-s + 0.990·19-s − 0.0519·21-s − 1.74·23-s − 0.991·25-s + 0.377·27-s + 0.678·29-s + 0.996·31-s + 0.0527·33-s + 0.0241·35-s − 1.68·37-s − 0.0533·39-s − 0.563·41-s + 1.16·43-s − 0.0861·45-s − 1.95·47-s − 0.927·49-s − 0.255·51-s − 0.414·53-s − 0.0245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 93 T + p^{3} T^{2} \) |
| 19 | \( 1 - 82 T + p^{3} T^{2} \) |
| 23 | \( 1 + 192 T + p^{3} T^{2} \) |
| 29 | \( 1 - 106 T + p^{3} T^{2} \) |
| 31 | \( 1 - 172 T + p^{3} T^{2} \) |
| 37 | \( 1 + 379 T + p^{3} T^{2} \) |
| 41 | \( 1 + 148 T + p^{3} T^{2} \) |
| 43 | \( 1 - 329 T + p^{3} T^{2} \) |
| 47 | \( 1 + 631 T + p^{3} T^{2} \) |
| 53 | \( 1 + 160 T + p^{3} T^{2} \) |
| 59 | \( 1 - 478 T + p^{3} T^{2} \) |
| 61 | \( 1 + 300 T + p^{3} T^{2} \) |
| 67 | \( 1 - 722 T + p^{3} T^{2} \) |
| 71 | \( 1 - 335 T + p^{3} T^{2} \) |
| 73 | \( 1 - 90 T + p^{3} T^{2} \) |
| 79 | \( 1 + 788 T + p^{3} T^{2} \) |
| 83 | \( 1 + 96 T + p^{3} T^{2} \) |
| 89 | \( 1 + 866 T + p^{3} T^{2} \) |
| 97 | \( 1 + 998 T + p^{3} T^{2} \) |
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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
−9.613704309540777250268596589595, −8.167813759301661801183046517703, −8.088289268946141994624778713912, −6.70024482664581999822823449629, −5.74063915498051990984800332550, −5.16713873729953930343733992984, −3.80408435418353289430383969977, −2.82707237502241495176358534698, −1.46130232553440277149403682400, 0,
1.46130232553440277149403682400, 2.82707237502241495176358534698, 3.80408435418353289430383969977, 5.16713873729953930343733992984, 5.74063915498051990984800332550, 6.70024482664581999822823449629, 8.088289268946141994624778713912, 8.167813759301661801183046517703, 9.613704309540777250268596589595
