| L(s) = 1 | + 3-s + 2·5-s + 9-s + 11-s − 2·13-s + 2·15-s + 6·17-s + 4·23-s − 25-s + 27-s − 2·29-s + 33-s + 10·37-s − 2·39-s − 6·41-s − 8·43-s + 2·45-s − 4·47-s − 7·49-s + 6·51-s + 6·53-s + 2·55-s + 12·59-s + 2·61-s − 4·65-s − 4·67-s + 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.516·15-s + 1.45·17-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.174·33-s + 1.64·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.298·45-s − 0.583·47-s − 49-s + 0.840·51-s + 0.824·53-s + 0.269·55-s + 1.56·59-s + 0.256·61-s − 0.496·65-s − 0.488·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.18679994234908, −13.38386091239690, −13.22679169148604, −12.78846467412685, −12.08079211479514, −11.58722129042858, −11.24405437207594, −10.22091985670385, −10.02715374821653, −9.755224463692543, −9.119626990505621, −8.566380634877525, −8.140257243519747, −7.398076267352360, −7.135146118951769, −6.421358642979998, −5.806416641792556, −5.421344679447693, −4.778163011338439, −4.166748988013049, −3.431823349868826, −2.925880578941792, −2.391623517301234, −1.527204903762651, −1.226808060436726, 0,
1.226808060436726, 1.527204903762651, 2.391623517301234, 2.925880578941792, 3.431823349868826, 4.166748988013049, 4.778163011338439, 5.421344679447693, 5.806416641792556, 6.421358642979998, 7.135146118951769, 7.398076267352360, 8.140257243519747, 8.566380634877525, 9.119626990505621, 9.755224463692543, 10.02715374821653, 10.22091985670385, 11.24405437207594, 11.58722129042858, 12.08079211479514, 12.78846467412685, 13.22679169148604, 13.38386091239690, 14.18679994234908
