| L(s) = 1 | − 4·5-s + 2·7-s + 4·11-s − 2·13-s + 2·17-s − 8·19-s − 4·23-s + 11·25-s − 6·31-s − 8·35-s + 2·37-s − 6·41-s − 4·47-s − 3·49-s − 16·55-s − 4·59-s − 14·61-s + 8·65-s − 4·67-s − 12·71-s − 10·73-s + 8·77-s + 10·79-s − 12·83-s − 8·85-s + 14·89-s − 4·91-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.755·7-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 1.83·19-s − 0.834·23-s + 11/5·25-s − 1.07·31-s − 1.35·35-s + 0.328·37-s − 0.937·41-s − 0.583·47-s − 3/7·49-s − 2.15·55-s − 0.520·59-s − 1.79·61-s + 0.992·65-s − 0.488·67-s − 1.42·71-s − 1.17·73-s + 0.911·77-s + 1.12·79-s − 1.31·83-s − 0.867·85-s + 1.48·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.164972285247004052157920295794, −8.432951706216498136071721435997, −7.80892757070983972668727076655, −7.08289458029334266024179255426, −6.13258408514753038234534207414, −4.69737181089599911757008902155, −4.20448032364885463054764539919, −3.32102035102375871114821653214, −1.70795903707382777058669654679, 0,
1.70795903707382777058669654679, 3.32102035102375871114821653214, 4.20448032364885463054764539919, 4.69737181089599911757008902155, 6.13258408514753038234534207414, 7.08289458029334266024179255426, 7.80892757070983972668727076655, 8.432951706216498136071721435997, 9.164972285247004052157920295794
