| L(s) = 1 | − 3-s + 7-s + 9-s − 6·13-s − 2·17-s − 8·19-s − 21-s + 8·23-s − 27-s + 2·29-s − 4·31-s − 2·37-s + 6·39-s − 6·41-s − 4·43-s + 8·47-s + 49-s + 2·51-s + 10·53-s + 8·57-s + 4·59-s + 2·61-s + 63-s − 4·67-s − 8·69-s + 12·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.485·17-s − 1.83·19-s − 0.218·21-s + 1.66·23-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.328·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 1.37·53-s + 1.05·57-s + 0.520·59-s + 0.256·61-s + 0.125·63-s − 0.488·67-s − 0.963·69-s + 1.42·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.20672704080722, −14.82249309389605, −14.37409594632568, −13.60970991635523, −12.99183213573387, −12.64321206154807, −12.12721385459349, −11.51664258190316, −11.06648648416368, −10.40439559433984, −10.16884531464083, −9.346147270864579, −8.731162437456864, −8.399916829930757, −7.308901916425215, −7.199359593108802, −6.539739701373675, −5.855834845229384, −5.023625000913448, −4.849027693207710, −4.153726246746761, −3.332475088425546, −2.342010386795280, −2.029674829972114, −0.8502949742382285, 0,
0.8502949742382285, 2.029674829972114, 2.342010386795280, 3.332475088425546, 4.153726246746761, 4.849027693207710, 5.023625000913448, 5.855834845229384, 6.539739701373675, 7.199359593108802, 7.308901916425215, 8.399916829930757, 8.731162437456864, 9.346147270864579, 10.16884531464083, 10.40439559433984, 11.06648648416368, 11.51664258190316, 12.12721385459349, 12.64321206154807, 12.99183213573387, 13.60970991635523, 14.37409594632568, 14.82249309389605, 15.20672704080722
