| L(s) = 1 | + 2·3-s − 5-s + 9-s − 4·11-s − 2·13-s − 2·15-s − 4·19-s + 4·23-s + 25-s − 4·27-s − 29-s + 4·31-s − 8·33-s − 8·37-s − 4·39-s − 2·41-s − 2·43-s − 45-s − 2·47-s − 7·49-s − 14·53-s + 4·55-s − 8·57-s − 4·59-s + 2·61-s + 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.769·27-s − 0.185·29-s + 0.718·31-s − 1.39·33-s − 1.31·37-s − 0.640·39-s − 0.312·41-s − 0.304·43-s − 0.149·45-s − 0.291·47-s − 49-s − 1.92·53-s + 0.539·55-s − 1.05·57-s − 0.520·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.447815823351248624255569666132, −8.027408562704202386051135044891, −7.31681710275634255481737151107, −6.44472276226677034914841864912, −5.25063137109869661659534636131, −4.56486313062521434548978728557, −3.42132196256514672991252812611, −2.82213750841341656222084169597, −1.88960029593633480327087792402, 0,
1.88960029593633480327087792402, 2.82213750841341656222084169597, 3.42132196256514672991252812611, 4.56486313062521434548978728557, 5.25063137109869661659534636131, 6.44472276226677034914841864912, 7.31681710275634255481737151107, 8.027408562704202386051135044891, 8.447815823351248624255569666132
