L(s) = 1 | − 2·5-s + 4·11-s − 2·13-s − 4·19-s − 8·23-s − 25-s + 6·29-s − 8·31-s − 6·37-s − 6·41-s + 4·43-s − 7·49-s + 2·53-s − 8·55-s − 4·59-s + 2·61-s + 4·65-s − 4·67-s + 8·71-s − 10·73-s + 8·79-s + 4·83-s + 6·89-s + 8·95-s − 2·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 1.20·11-s − 0.554·13-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 49-s + 0.274·53-s − 1.07·55-s − 0.520·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.900·79-s + 0.439·83-s + 0.635·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9442090593\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9442090593\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.64599675985093, −15.05829452673096, −14.48918452319773, −14.15389235273326, −13.51491307337922, −12.67765959210174, −12.24267621333382, −11.83568449614213, −11.37355651264703, −10.63069471821061, −10.10734802687685, −9.483973044284615, −8.783652521615910, −8.368271646560552, −7.664988719159881, −7.181555997647955, −6.427031152086684, −6.031383441284441, −5.042770400000689, −4.427648396728269, −3.802185611069527, −3.396382727410840, −2.231132575131101, −1.628576146038598, −0.3906493503781686,
0.3906493503781686, 1.628576146038598, 2.231132575131101, 3.396382727410840, 3.802185611069527, 4.427648396728269, 5.042770400000689, 6.031383441284441, 6.427031152086684, 7.181555997647955, 7.664988719159881, 8.368271646560552, 8.783652521615910, 9.483973044284615, 10.10734802687685, 10.63069471821061, 11.37355651264703, 11.83568449614213, 12.24267621333382, 12.67765959210174, 13.51491307337922, 14.15389235273326, 14.48918452319773, 15.05829452673096, 15.64599675985093