| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 4·7-s − 8-s + 9-s − 11-s − 12-s − 2·13-s − 4·14-s + 16-s + 3·17-s − 18-s − 4·19-s − 4·21-s + 22-s + 24-s + 2·26-s − 27-s + 4·28-s − 4·29-s + 2·31-s − 32-s + 33-s − 3·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.917·19-s − 0.872·21-s + 0.213·22-s + 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.755·28-s − 0.742·29-s + 0.359·31-s − 0.176·32-s + 0.174·33-s − 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−14.34605752393957, −13.95952767650552, −13.05492787249077, −12.67300836729626, −12.09334374187646, −11.70132372357041, −11.08255593158430, −10.89784301151798, −10.27904990102271, −9.847511987581185, −9.202418206714428, −8.623345427065284, −8.110210898742350, −7.701312776528690, −7.244576866351241, −6.637857612817322, −5.953115517386672, −5.359202347277750, −5.036828924893922, −4.286446916857890, −3.794002403596218, −2.736114966652078, −2.184038029245981, −1.547173958200671, −0.8980522112762563, 0,
0.8980522112762563, 1.547173958200671, 2.184038029245981, 2.736114966652078, 3.794002403596218, 4.286446916857890, 5.036828924893922, 5.359202347277750, 5.953115517386672, 6.637857612817322, 7.244576866351241, 7.701312776528690, 8.110210898742350, 8.623345427065284, 9.202418206714428, 9.847511987581185, 10.27904990102271, 10.89784301151798, 11.08255593158430, 11.70132372357041, 12.09334374187646, 12.67300836729626, 13.05492787249077, 13.95952767650552, 14.34605752393957
