| L(s) = 1 | − 2·2-s + 2·4-s + 5-s + 3·7-s − 2·10-s − 5·11-s + 13-s − 6·14-s − 4·16-s + 2·19-s + 2·20-s + 10·22-s − 23-s + 25-s − 2·26-s + 6·28-s + 10·29-s + 2·31-s + 8·32-s + 3·35-s + 3·37-s − 4·38-s − 9·41-s − 4·43-s − 10·44-s + 2·46-s − 10·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s + 1.13·7-s − 0.632·10-s − 1.50·11-s + 0.277·13-s − 1.60·14-s − 16-s + 0.458·19-s + 0.447·20-s + 2.13·22-s − 0.208·23-s + 1/5·25-s − 0.392·26-s + 1.13·28-s + 1.85·29-s + 0.359·31-s + 1.41·32-s + 0.507·35-s + 0.493·37-s − 0.648·38-s − 1.40·41-s − 0.609·43-s − 1.50·44-s + 0.294·46-s − 1.45·47-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
| \(L(1)\) |
\(\approx\) |
\(1.314625900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.314625900\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.23764201417959, −12.82374594225671, −12.06929528740822, −11.56366833795984, −11.23477752932171, −10.64908853270225, −10.26553943486036, −9.981107105811277, −9.464753601739223, −8.818975112329332, −8.260986841167216, −8.062035147055409, −7.836129339417791, −6.959243175891872, −6.656700401580762, −5.963530865750501, −5.230694800131958, −4.808668594302607, −4.582882679832845, −3.441553787539279, −2.885677052840079, −2.164314039718959, −1.783847593627246, −1.082822052639379, −0.4689294424650835,
0.4689294424650835, 1.082822052639379, 1.783847593627246, 2.164314039718959, 2.885677052840079, 3.441553787539279, 4.582882679832845, 4.808668594302607, 5.230694800131958, 5.963530865750501, 6.656700401580762, 6.959243175891872, 7.836129339417791, 8.062035147055409, 8.260986841167216, 8.818975112329332, 9.464753601739223, 9.981107105811277, 10.26553943486036, 10.64908853270225, 11.23477752932171, 11.56366833795984, 12.06929528740822, 12.82374594225671, 13.23764201417959
