| L(s) = 1 | + 5-s + 4·7-s − 6·11-s − 13-s − 2·19-s + 23-s + 25-s − 9·29-s − 5·31-s + 4·35-s + 2·37-s + 9·41-s + 4·43-s − 3·47-s + 9·49-s + 6·53-s − 6·55-s + 2·61-s − 65-s + 10·67-s − 3·71-s − 7·73-s − 24·77-s + 10·79-s − 12·83-s − 4·91-s − 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.80·11-s − 0.277·13-s − 0.458·19-s + 0.208·23-s + 1/5·25-s − 1.67·29-s − 0.898·31-s + 0.676·35-s + 0.328·37-s + 1.40·41-s + 0.609·43-s − 0.437·47-s + 9/7·49-s + 0.824·53-s − 0.809·55-s + 0.256·61-s − 0.124·65-s + 1.22·67-s − 0.356·71-s − 0.819·73-s − 2.73·77-s + 1.12·79-s − 1.31·83-s − 0.419·91-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.24461948385105, −15.50946554624930, −15.06923843383268, −14.47415445357172, −14.18808408028271, −13.21558354874262, −13.04488306199608, −12.44152819024194, −11.54360727382948, −11.06045033749230, −10.72255826136647, −10.09324469728058, −9.376214618673033, −8.737922981950419, −8.110972905743438, −7.544278107820017, −7.270735433119548, −6.094277680218076, −5.481898035250026, −5.113039300690297, −4.453836138437037, −3.632473646596222, −2.444836347847677, −2.229535192620314, −1.234332879429931, 0,
1.234332879429931, 2.229535192620314, 2.444836347847677, 3.632473646596222, 4.453836138437037, 5.113039300690297, 5.481898035250026, 6.094277680218076, 7.270735433119548, 7.544278107820017, 8.110972905743438, 8.737922981950419, 9.376214618673033, 10.09324469728058, 10.72255826136647, 11.06045033749230, 11.54360727382948, 12.44152819024194, 13.04488306199608, 13.21558354874262, 14.18808408028271, 14.47415445357172, 15.06923843383268, 15.50946554624930, 16.24461948385105
