| L(s) = 1 | + 5-s + 6·13-s + 2·17-s − 8·19-s + 8·23-s + 25-s + 2·29-s + 4·31-s − 2·37-s − 6·41-s − 4·43-s − 8·47-s − 10·53-s − 4·59-s + 2·61-s + 6·65-s − 4·67-s − 12·71-s + 2·73-s − 8·79-s + 4·83-s + 2·85-s − 6·89-s − 8·95-s + 18·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.66·13-s + 0.485·17-s − 1.83·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s + 0.718·31-s − 0.328·37-s − 0.937·41-s − 0.609·43-s − 1.16·47-s − 1.37·53-s − 0.520·59-s + 0.256·61-s + 0.744·65-s − 0.488·67-s − 1.42·71-s + 0.234·73-s − 0.900·79-s + 0.439·83-s + 0.216·85-s − 0.635·89-s − 0.820·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.10363397221841, −14.75236126356596, −14.14270096972825, −13.51099791104197, −13.08103411578765, −12.84236937219487, −12.00352298322709, −11.47543535113648, −10.81111678594450, −10.57869145783334, −9.956936808548619, −9.218403099402947, −8.684331810752629, −8.382912128583104, −7.721343638015191, −6.748187604779745, −6.494468128186026, −5.994714868141580, −5.178245458019959, −4.676186795527849, −3.922171030127764, −3.244732430287553, −2.678088375266950, −1.618769259174455, −1.238054441997853, 0,
1.238054441997853, 1.618769259174455, 2.678088375266950, 3.244732430287553, 3.922171030127764, 4.676186795527849, 5.178245458019959, 5.994714868141580, 6.494468128186026, 6.748187604779745, 7.721343638015191, 8.382912128583104, 8.684331810752629, 9.218403099402947, 9.956936808548619, 10.57869145783334, 10.81111678594450, 11.47543535113648, 12.00352298322709, 12.84236937219487, 13.08103411578765, 13.51099791104197, 14.14270096972825, 14.75236126356596, 15.10363397221841
