| L(s) = 1 | + 2·3-s − 7-s + 9-s − 4·11-s − 6·13-s − 2·21-s − 5·25-s − 4·27-s − 2·29-s − 2·31-s − 8·33-s + 2·37-s − 12·39-s − 6·41-s − 4·43-s + 2·47-s + 49-s + 14·53-s + 14·59-s + 12·61-s − 63-s + 4·67-s − 2·73-s − 10·75-s + 4·77-s − 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.436·21-s − 25-s − 0.769·27-s − 0.371·29-s − 0.359·31-s − 1.39·33-s + 0.328·37-s − 1.92·39-s − 0.937·41-s − 0.609·43-s + 0.291·47-s + 1/7·49-s + 1.92·53-s + 1.82·59-s + 1.53·61-s − 0.125·63-s + 0.488·67-s − 0.234·73-s − 1.15·75-s + 0.455·77-s − 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.28741495891685, −12.93105330207440, −12.17494368036178, −11.82953958614367, −11.40078084043493, −10.62819483341527, −10.14157206322340, −9.882706384839230, −9.449158240504855, −8.922612777623247, −8.317690813835942, −8.076397159512058, −7.504676522559195, −7.110669420475242, −6.730613436122623, −5.702171081280501, −5.506477918301343, −4.970214708572574, −4.265838173508428, −3.709590932617703, −3.253583017120013, −2.490124815653340, −2.378792588890073, −1.836486044936118, −0.6497027917061186, 0,
0.6497027917061186, 1.836486044936118, 2.378792588890073, 2.490124815653340, 3.253583017120013, 3.709590932617703, 4.265838173508428, 4.970214708572574, 5.506477918301343, 5.702171081280501, 6.730613436122623, 7.110669420475242, 7.504676522559195, 8.076397159512058, 8.317690813835942, 8.922612777623247, 9.449158240504855, 9.882706384839230, 10.14157206322340, 10.62819483341527, 11.40078084043493, 11.82953958614367, 12.17494368036178, 12.93105330207440, 13.28741495891685
