| L(s) = 1 | + 2-s + 2.73·3-s + 4-s + 5-s + 2.73·6-s − 7-s + 8-s + 4.46·9-s + 10-s + 2.73·12-s + 1.46·13-s − 14-s + 2.73·15-s + 16-s − 3.46·17-s + 4.46·18-s + 2.73·19-s + 20-s − 2.73·21-s + 1.26·23-s + 2.73·24-s + 25-s + 1.46·26-s + 3.99·27-s − 28-s + 4.73·29-s + 2.73·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.57·3-s + 0.5·4-s + 0.447·5-s + 1.11·6-s − 0.377·7-s + 0.353·8-s + 1.48·9-s + 0.316·10-s + 0.788·12-s + 0.406·13-s − 0.267·14-s + 0.705·15-s + 0.250·16-s − 0.840·17-s + 1.05·18-s + 0.626·19-s + 0.223·20-s − 0.596·21-s + 0.264·23-s + 0.557·24-s + 0.200·25-s + 0.287·26-s + 0.769·27-s − 0.188·28-s + 0.878·29-s + 0.498·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.920817964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.920817964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2.73T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 4.73T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 6.73T + 79T^{2} \) |
| 83 | \( 1 - 4.39T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 5.80T + 97T^{2} \) |
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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
−7.78059940453279336224313384147, −7.15332615899911342776755347434, −6.36523581560699696948527877599, −5.85179520421783260878237406887, −4.63933161467128895793958257258, −4.29422184733729054405661842238, −3.19090855686503637223716267846, −2.89984070631018003729540512118, −2.11732139466118746846439438010, −1.14327275344601601510968457782,
1.14327275344601601510968457782, 2.11732139466118746846439438010, 2.89984070631018003729540512118, 3.19090855686503637223716267846, 4.29422184733729054405661842238, 4.63933161467128895793958257258, 5.85179520421783260878237406887, 6.36523581560699696948527877599, 7.15332615899911342776755347434, 7.78059940453279336224313384147
