L(s) = 1 | + 5-s − 2·7-s + 2·13-s + 6·17-s − 8·19-s + 6·23-s + 25-s + 6·29-s + 4·31-s − 2·35-s + 8·37-s − 6·41-s − 43-s + 6·47-s − 3·49-s + 12·53-s − 4·61-s + 2·65-s + 4·67-s − 10·73-s − 8·79-s + 6·85-s − 6·89-s − 4·91-s − 8·95-s + 14·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.554·13-s + 1.45·17-s − 1.83·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.338·35-s + 1.31·37-s − 0.937·41-s − 0.152·43-s + 0.875·47-s − 3/7·49-s + 1.64·53-s − 0.512·61-s + 0.248·65-s + 0.488·67-s − 1.17·73-s − 0.900·79-s + 0.650·85-s − 0.635·89-s − 0.419·91-s − 0.820·95-s + 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.543855233\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.543855233\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.09612493936237, −14.58252694137720, −14.06050346072008, −13.37503935119456, −13.03286862798209, −12.57106496528278, −11.97240580892019, −11.39209216046349, −10.63616461398223, −10.25991555809503, −9.846062737537062, −9.120620830473981, −8.573493782198921, −8.185751063448940, −7.276580055452654, −6.791011284801255, −6.094047395002959, −5.874037076494148, −4.941541667291374, −4.380768707443090, −3.585651935712347, −2.959531084035998, −2.377799801488345, −1.355003125574179, −0.6522814720379540,
0.6522814720379540, 1.355003125574179, 2.377799801488345, 2.959531084035998, 3.585651935712347, 4.380768707443090, 4.941541667291374, 5.874037076494148, 6.094047395002959, 6.791011284801255, 7.276580055452654, 8.185751063448940, 8.573493782198921, 9.120620830473981, 9.846062737537062, 10.25991555809503, 10.63616461398223, 11.39209216046349, 11.97240580892019, 12.57106496528278, 13.03286862798209, 13.37503935119456, 14.06050346072008, 14.58252694137720, 15.09612493936237