| L(s) = 1 | − 5-s + 5·11-s + 13-s − 3·17-s + 3·23-s + 25-s + 6·29-s + 2·31-s − 3·37-s − 41-s + 2·43-s − 6·47-s + 10·53-s − 5·55-s − 59-s + 2·61-s − 65-s + 13·67-s − 11·73-s − 5·79-s + 2·83-s + 3·85-s + 89-s − 5·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.50·11-s + 0.277·13-s − 0.727·17-s + 0.625·23-s + 1/5·25-s + 1.11·29-s + 0.359·31-s − 0.493·37-s − 0.156·41-s + 0.304·43-s − 0.875·47-s + 1.37·53-s − 0.674·55-s − 0.130·59-s + 0.256·61-s − 0.124·65-s + 1.58·67-s − 1.28·73-s − 0.562·79-s + 0.219·83-s + 0.325·85-s + 0.105·89-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.18282833463806, −12.64866894069602, −12.09323933301547, −11.85015404740657, −11.31354063484942, −10.97591124213490, −10.41148237997158, −9.851611232696755, −9.417195772357357, −8.813129986784562, −8.569416599233115, −8.124014900128465, −7.363058747103814, −6.891146605392180, −6.592398666130338, −6.129277691027596, −5.403078439559003, −4.893610262602373, −4.229649924819076, −4.003256124741693, −3.339539856631970, −2.783680921995459, −2.109296340989306, −1.322693555985191, −0.9248529314903778, 0,
0.9248529314903778, 1.322693555985191, 2.109296340989306, 2.783680921995459, 3.339539856631970, 4.003256124741693, 4.229649924819076, 4.893610262602373, 5.403078439559003, 6.129277691027596, 6.592398666130338, 6.891146605392180, 7.363058747103814, 8.124014900128465, 8.569416599233115, 8.813129986784562, 9.417195772357357, 9.851611232696755, 10.41148237997158, 10.97591124213490, 11.31354063484942, 11.85015404740657, 12.09323933301547, 12.64866894069602, 13.18282833463806
