L(s) = 1 | + 3-s + 7-s + 9-s − 11-s + 2·13-s + 2·17-s + 21-s − 4·23-s + 27-s − 6·29-s − 4·31-s − 33-s − 10·37-s + 2·39-s − 2·41-s + 4·43-s + 8·47-s + 49-s + 2·51-s + 2·53-s + 4·59-s + 10·61-s + 63-s − 4·67-s − 4·69-s − 8·71-s + 10·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.485·17-s + 0.218·21-s − 0.834·23-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.174·33-s − 1.64·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s + 0.520·59-s + 1.28·61-s + 0.125·63-s − 0.488·67-s − 0.481·69-s − 0.949·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.637690724\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.637690724\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.44018278084186, −12.13500735409567, −11.63027946987214, −11.04551383847462, −10.73328745462871, −10.20028098731304, −9.830389927791899, −9.241192456832830, −8.795460613272624, −8.484536097372055, −7.855269579646747, −7.608214693401848, −7.018721681059886, −6.628923541378298, −5.860320393687580, −5.436780681303918, −5.191763874889396, −4.258994979861210, −3.936207576618538, −3.529049601713302, −2.869229511110328, −2.269932190698050, −1.767604973507826, −1.249666720888109, −0.3954933939254357,
0.3954933939254357, 1.249666720888109, 1.767604973507826, 2.269932190698050, 2.869229511110328, 3.529049601713302, 3.936207576618538, 4.258994979861210, 5.191763874889396, 5.436780681303918, 5.860320393687580, 6.628923541378298, 7.018721681059886, 7.608214693401848, 7.855269579646747, 8.484536097372055, 8.795460613272624, 9.241192456832830, 9.830389927791899, 10.20028098731304, 10.73328745462871, 11.04551383847462, 11.63027946987214, 12.13500735409567, 12.44018278084186