| L(s) = 1 | − 2·4-s + 5-s + 7-s − 3·11-s + 13-s + 4·16-s − 4·19-s − 2·20-s − 9·23-s + 25-s − 2·28-s − 6·29-s − 2·31-s + 35-s + 37-s − 3·41-s + 2·43-s + 6·44-s + 6·47-s − 6·49-s − 2·52-s − 9·53-s − 3·55-s + 12·59-s − 5·61-s − 8·64-s + 65-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.377·7-s − 0.904·11-s + 0.277·13-s + 16-s − 0.917·19-s − 0.447·20-s − 1.87·23-s + 1/5·25-s − 0.377·28-s − 1.11·29-s − 0.359·31-s + 0.169·35-s + 0.164·37-s − 0.468·41-s + 0.304·43-s + 0.904·44-s + 0.875·47-s − 6/7·49-s − 0.277·52-s − 1.23·53-s − 0.404·55-s + 1.56·59-s − 0.640·61-s − 64-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.46610946946171, −13.00765764813291, −12.62265186049224, −12.25339782120383, −11.48224550375148, −11.06370752721991, −10.52859719612303, −10.00779056903724, −9.798417071692608, −9.123118695244988, −8.609372349086001, −8.299272793996002, −7.690810310448815, −7.415948030395128, −6.460873041162434, −6.028654772194932, −5.539582887830519, −5.141713857039384, −4.432368115776646, −4.115785075776668, −3.481506393204000, −2.824221855655002, −1.950401136108165, −1.745536172424063, −0.6401583267080229, 0,
0.6401583267080229, 1.745536172424063, 1.950401136108165, 2.824221855655002, 3.481506393204000, 4.115785075776668, 4.432368115776646, 5.141713857039384, 5.539582887830519, 6.028654772194932, 6.460873041162434, 7.415948030395128, 7.690810310448815, 8.299272793996002, 8.609372349086001, 9.123118695244988, 9.798417071692608, 10.00779056903724, 10.52859719612303, 11.06370752721991, 11.48224550375148, 12.25339782120383, 12.62265186049224, 13.00765764813291, 13.46610946946171
