| L(s) = 1 | + 3-s − 2·4-s + 9-s − 11-s − 2·12-s − 13-s + 4·16-s − 17-s + 4·19-s + 3·23-s + 27-s − 8·29-s + 4·31-s − 33-s − 2·36-s − 3·37-s − 39-s + 9·41-s + 8·43-s + 2·44-s + 10·47-s + 4·48-s − 51-s + 2·52-s + 53-s + 4·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 0.277·13-s + 16-s − 0.242·17-s + 0.917·19-s + 0.625·23-s + 0.192·27-s − 1.48·29-s + 0.718·31-s − 0.174·33-s − 1/3·36-s − 0.493·37-s − 0.160·39-s + 1.40·41-s + 1.21·43-s + 0.301·44-s + 1.45·47-s + 0.577·48-s − 0.140·51-s + 0.277·52-s + 0.137·53-s + 0.529·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.243890210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.243890210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−14.45007890085594, −14.09114649787233, −13.47904078922344, −13.26139828912245, −12.48885749164283, −12.31401749472529, −11.46033311984541, −10.81947143004208, −10.42240120891292, −9.597121443400266, −9.333382560035357, −9.002908787507629, −8.193731495316785, −7.808263919225055, −7.307175550214693, −6.663423148900037, −5.727221634490123, −5.368973882751142, −4.758903732526453, −3.999324741584385, −3.702831238913922, −2.805500511572452, −2.307559669252606, −1.238781489975112, −0.5731099360654725,
0.5731099360654725, 1.238781489975112, 2.307559669252606, 2.805500511572452, 3.702831238913922, 3.999324741584385, 4.758903732526453, 5.368973882751142, 5.727221634490123, 6.663423148900037, 7.307175550214693, 7.808263919225055, 8.193731495316785, 9.002908787507629, 9.333382560035357, 9.597121443400266, 10.42240120891292, 10.81947143004208, 11.46033311984541, 12.31401749472529, 12.48885749164283, 13.26139828912245, 13.47904078922344, 14.09114649787233, 14.45007890085594
