| L(s) = 1 | − 5-s + 7-s − 4·11-s + 6·13-s − 2·17-s + 25-s + 6·29-s − 8·31-s − 35-s + 10·37-s − 2·41-s + 4·43-s + 8·47-s + 49-s − 2·53-s + 4·55-s + 8·59-s + 14·61-s − 6·65-s − 12·67-s − 16·71-s + 2·73-s − 4·77-s + 8·79-s − 8·83-s + 2·85-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.169·35-s + 1.64·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.274·53-s + 0.539·55-s + 1.04·59-s + 1.79·61-s − 0.744·65-s − 1.46·67-s − 1.89·71-s + 0.234·73-s − 0.455·77-s + 0.900·79-s − 0.878·83-s + 0.216·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.927428926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927428926\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.88016369319790, −15.01771537465767, −14.76988039237213, −13.91343919967315, −13.38073397908890, −13.06937818598680, −12.39860890472308, −11.74212023560248, −10.98473202505474, −10.94764057270331, −10.24894809720647, −9.480789773470563, −8.706146607095536, −8.406320210615155, −7.786807891388378, −7.205663305690906, −6.496750816642591, −5.734556447802273, −5.348872896428446, −4.340807955670952, −4.033690262381856, −3.075135529630030, −2.469247279320023, −1.476733243231201, −0.5956583316240391,
0.5956583316240391, 1.476733243231201, 2.469247279320023, 3.075135529630030, 4.033690262381856, 4.340807955670952, 5.348872896428446, 5.734556447802273, 6.496750816642591, 7.205663305690906, 7.786807891388378, 8.406320210615155, 8.706146607095536, 9.480789773470563, 10.24894809720647, 10.94764057270331, 10.98473202505474, 11.74212023560248, 12.39860890472308, 13.06937818598680, 13.38073397908890, 13.91343919967315, 14.76988039237213, 15.01771537465767, 15.88016369319790
