| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 2·11-s + 2·13-s + 2·14-s + 16-s + 4·19-s + 2·22-s + 4·23-s − 2·26-s − 2·28-s + 6·29-s − 32-s + 10·37-s − 4·38-s − 2·41-s + 4·43-s − 2·44-s − 4·46-s + 8·47-s − 3·49-s + 2·52-s + 2·53-s + 2·56-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.603·11-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.917·19-s + 0.426·22-s + 0.834·23-s − 0.392·26-s − 0.377·28-s + 1.11·29-s − 0.176·32-s + 1.64·37-s − 0.648·38-s − 0.312·41-s + 0.609·43-s − 0.301·44-s − 0.589·46-s + 1.16·47-s − 3/7·49-s + 0.277·52-s + 0.274·53-s + 0.267·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.31128185056241, −13.69390089006357, −13.33594187897190, −12.80317212101968, −12.27200613144141, −11.79926705972207, −11.15626714282017, −10.78912751127997, −10.19368709667589, −9.812891749991187, −9.209397087189867, −8.854924593473505, −8.235448255892189, −7.597844311368922, −7.322911390367875, −6.536174427659912, −6.170400521156252, −5.574817073339641, −4.933484706125071, −4.225759512733907, −3.508805490519659, −2.748663638289470, −2.641074389313036, −1.402919705490135, −0.9148682085326425, 0,
0.9148682085326425, 1.402919705490135, 2.641074389313036, 2.748663638289470, 3.508805490519659, 4.225759512733907, 4.933484706125071, 5.574817073339641, 6.170400521156252, 6.536174427659912, 7.322911390367875, 7.597844311368922, 8.235448255892189, 8.854924593473505, 9.209397087189867, 9.812891749991187, 10.19368709667589, 10.78912751127997, 11.15626714282017, 11.79926705972207, 12.27200613144141, 12.80317212101968, 13.33594187897190, 13.69390089006357, 14.31128185056241
