| L(s) = 1 | − 5-s − 2·17-s + 4·19-s − 4·23-s + 25-s + 6·29-s − 8·31-s + 6·37-s − 2·41-s + 4·43-s − 7·49-s − 6·53-s − 2·61-s − 8·67-s − 6·73-s − 4·79-s − 12·83-s + 2·85-s + 6·89-s − 4·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 4·115-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.986·37-s − 0.312·41-s + 0.609·43-s − 49-s − 0.824·53-s − 0.256·61-s − 0.977·67-s − 0.702·73-s − 0.450·79-s − 1.31·83-s + 0.216·85-s + 0.635·89-s − 0.410·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.373·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.076613849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.076613849\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.85175738479565, −12.47150511272044, −11.75330728297025, −11.65785904518764, −11.05866261219600, −10.61798633806852, −10.06323809526552, −9.660784468312120, −9.091395095900965, −8.720871321060820, −8.125004336244346, −7.637881590462910, −7.355179103709929, −6.705806441696121, −6.161462616992758, −5.797210677424412, −5.042408472155337, −4.695782722835501, −4.071582663269667, −3.615600350189639, −2.969264952275757, −2.522766448786468, −1.701501786733309, −1.190127207554581, −0.2941746201152622,
0.2941746201152622, 1.190127207554581, 1.701501786733309, 2.522766448786468, 2.969264952275757, 3.615600350189639, 4.071582663269667, 4.695782722835501, 5.042408472155337, 5.797210677424412, 6.161462616992758, 6.705806441696121, 7.355179103709929, 7.637881590462910, 8.125004336244346, 8.720871321060820, 9.091395095900965, 9.660784468312120, 10.06323809526552, 10.61798633806852, 11.05866261219600, 11.65785904518764, 11.75330728297025, 12.47150511272044, 12.85175738479565
