| L(s) = 1 | − 3-s − 5-s + 9-s + 15-s − 6·17-s + 4·23-s + 25-s − 27-s + 10·29-s − 6·37-s − 2·41-s − 4·43-s − 45-s − 7·49-s + 6·51-s + 6·53-s − 6·61-s − 4·67-s − 4·69-s + 16·71-s + 2·73-s − 75-s + 81-s − 4·83-s + 6·85-s − 10·87-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.258·15-s − 1.45·17-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.986·37-s − 0.312·41-s − 0.609·43-s − 0.149·45-s − 49-s + 0.840·51-s + 0.824·53-s − 0.768·61-s − 0.488·67-s − 0.481·69-s + 1.89·71-s + 0.234·73-s − 0.115·75-s + 1/9·81-s − 0.439·83-s + 0.650·85-s − 1.07·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9785978032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9785978032\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.33744913656952, −12.65213911858753, −12.30038000713464, −11.83187794767798, −11.39493661219807, −10.78009202057624, −10.68039077994485, −9.987736128190862, −9.435633236183139, −8.940248153755269, −8.305093471474632, −8.164166679663510, −7.248819484418260, −6.782437086516097, −6.615155794466334, −5.950568211677417, −5.145282903013650, −4.913017573652334, −4.331835325448163, −3.804423896352827, −3.077988699983345, −2.568016321935599, −1.784740066770877, −1.106940987235856, −0.3281380948836063,
0.3281380948836063, 1.106940987235856, 1.784740066770877, 2.568016321935599, 3.077988699983345, 3.804423896352827, 4.331835325448163, 4.913017573652334, 5.145282903013650, 5.950568211677417, 6.615155794466334, 6.782437086516097, 7.248819484418260, 8.164166679663510, 8.305093471474632, 8.940248153755269, 9.435633236183139, 9.987736128190862, 10.68039077994485, 10.78009202057624, 11.39493661219807, 11.83187794767798, 12.30038000713464, 12.65213911858753, 13.33744913656952
