| L(s) = 1 | + 5-s + 2·7-s − 2·13-s − 2·17-s + 4·19-s − 2·23-s + 25-s + 2·29-s − 8·31-s + 2·35-s + 6·37-s − 10·41-s + 8·43-s − 6·47-s − 3·49-s − 2·53-s + 12·59-s + 2·61-s − 2·65-s + 4·67-s + 6·71-s − 10·79-s + 4·83-s − 2·85-s + 8·89-s − 4·91-s + 4·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.417·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.338·35-s + 0.986·37-s − 1.56·41-s + 1.21·43-s − 0.875·47-s − 3/7·49-s − 0.274·53-s + 1.56·59-s + 0.256·61-s − 0.248·65-s + 0.488·67-s + 0.712·71-s − 1.12·79-s + 0.439·83-s − 0.216·85-s + 0.847·89-s − 0.419·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−14.23631424169960, −13.64511722248858, −13.19688706568551, −12.73941340423306, −12.12253652348961, −11.63254390356478, −11.24305275844078, −10.69923954963418, −10.17117923109319, −9.519749185923041, −9.361141383653034, −8.542863842950585, −8.115534552511708, −7.615441934679426, −6.988518546708310, −6.588168453446429, −5.813979687480554, −5.297631008579294, −4.946971278204794, −4.245281560335544, −3.654648721059525, −2.900975529806704, −2.254217545727055, −1.710509582573470, −0.9923098296022502, 0,
0.9923098296022502, 1.710509582573470, 2.254217545727055, 2.900975529806704, 3.654648721059525, 4.245281560335544, 4.946971278204794, 5.297631008579294, 5.813979687480554, 6.588168453446429, 6.988518546708310, 7.615441934679426, 8.115534552511708, 8.542863842950585, 9.361141383653034, 9.519749185923041, 10.17117923109319, 10.69923954963418, 11.24305275844078, 11.63254390356478, 12.12253652348961, 12.73941340423306, 13.19688706568551, 13.64511722248858, 14.23631424169960
