| L(s) = 1 | − 84·5-s − 49·7-s − 336·11-s + 584·13-s + 1.45e3·17-s − 470·19-s − 4.20e3·23-s + 3.93e3·25-s − 4.86e3·29-s + 7.37e3·31-s + 4.11e3·35-s + 1.43e4·37-s − 6.22e3·41-s − 3.70e3·43-s − 1.81e3·47-s + 2.40e3·49-s + 3.72e4·53-s + 2.82e4·55-s + 3.43e4·59-s + 2.44e4·61-s − 4.90e4·65-s + 1.74e4·67-s + 2.82e4·71-s + 3.60e3·73-s + 1.64e4·77-s − 4.28e4·79-s − 3.52e4·83-s + ⋯ |
| L(s) = 1 | − 1.50·5-s − 0.377·7-s − 0.837·11-s + 0.958·13-s + 1.22·17-s − 0.298·19-s − 1.65·23-s + 1.25·25-s − 1.07·29-s + 1.37·31-s + 0.567·35-s + 1.72·37-s − 0.578·41-s − 0.305·43-s − 0.119·47-s + 1/7·49-s + 1.82·53-s + 1.25·55-s + 1.28·59-s + 0.842·61-s − 1.44·65-s + 0.474·67-s + 0.664·71-s + 0.0791·73-s + 0.316·77-s − 0.772·79-s − 0.560·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 84 T + p^{5} T^{2} \) |
| 11 | \( 1 + 336 T + p^{5} T^{2} \) |
| 13 | \( 1 - 584 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1458 T + p^{5} T^{2} \) |
| 19 | \( 1 + 470 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4200 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4866 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7372 T + p^{5} T^{2} \) |
| 37 | \( 1 - 14330 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6222 T + p^{5} T^{2} \) |
| 43 | \( 1 + 3704 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1812 T + p^{5} T^{2} \) |
| 53 | \( 1 - 37242 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34302 T + p^{5} T^{2} \) |
| 61 | \( 1 - 24476 T + p^{5} T^{2} \) |
| 67 | \( 1 - 17452 T + p^{5} T^{2} \) |
| 71 | \( 1 - 28224 T + p^{5} T^{2} \) |
| 73 | \( 1 - 3602 T + p^{5} T^{2} \) |
| 79 | \( 1 + 42872 T + p^{5} T^{2} \) |
| 83 | \( 1 + 35202 T + p^{5} T^{2} \) |
| 89 | \( 1 + 26730 T + p^{5} T^{2} \) |
| 97 | \( 1 + 16978 T + p^{5} T^{2} \) |
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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
−8.455991543562299140087365805345, −8.064379679033587172021135779660, −7.35133628722897078587299136547, −6.26125281948782645982018079250, −5.38008102236983872282531512132, −4.10940062884351550158318461156, −3.64446089737654101390556282665, −2.53041997259828291007848121843, −0.955067700511637292833880047392, 0,
0.955067700511637292833880047392, 2.53041997259828291007848121843, 3.64446089737654101390556282665, 4.10940062884351550158318461156, 5.38008102236983872282531512132, 6.26125281948782645982018079250, 7.35133628722897078587299136547, 8.064379679033587172021135779660, 8.455991543562299140087365805345
