| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s − 5·11-s − 3·13-s − 15-s − 5·17-s − 21-s + 23-s + 25-s + 5·27-s − 3·29-s − 6·31-s + 5·33-s + 35-s + 4·37-s + 3·39-s − 2·43-s − 2·45-s + 9·47-s + 49-s + 5·51-s + 6·53-s − 5·55-s − 6·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.50·11-s − 0.832·13-s − 0.258·15-s − 1.21·17-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 1.07·31-s + 0.870·33-s + 0.169·35-s + 0.657·37-s + 0.480·39-s − 0.304·43-s − 0.298·45-s + 1.31·47-s + 1/7·49-s + 0.700·51-s + 0.824·53-s − 0.674·55-s − 0.781·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.93012293387958, −14.01983028945835, −13.83367712755602, −13.16580486678280, −12.65083710515113, −12.30013470093353, −11.53798474503151, −11.07401575532783, −10.69493833684612, −10.32161660485455, −9.465478243716190, −9.120831930199229, −8.475866291941059, −7.830053368457308, −7.431743430145102, −6.731221696615358, −6.132997199853179, −5.527238617586563, −5.104814112209160, −4.726893102740679, −3.869836681051432, −2.971890022904718, −2.359001523398222, −1.991902601503630, −0.7191896363595036, 0,
0.7191896363595036, 1.991902601503630, 2.359001523398222, 2.971890022904718, 3.869836681051432, 4.726893102740679, 5.104814112209160, 5.527238617586563, 6.132997199853179, 6.731221696615358, 7.431743430145102, 7.830053368457308, 8.475866291941059, 9.120831930199229, 9.465478243716190, 10.32161660485455, 10.69493833684612, 11.07401575532783, 11.53798474503151, 12.30013470093353, 12.65083710515113, 13.16580486678280, 13.83367712755602, 14.01983028945835, 14.93012293387958
