| L(s) = 1 | + 2-s + 4-s + 8-s − 2·11-s + 13-s + 16-s − 5·17-s + 5·19-s − 2·22-s − 4·23-s + 26-s + 2·29-s + 4·31-s + 32-s − 5·34-s + 4·37-s + 5·38-s + 3·41-s + 11·43-s − 2·44-s − 4·46-s + 2·47-s + 52-s − 6·53-s + 2·58-s + 6·59-s + 4·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s + 0.277·13-s + 1/4·16-s − 1.21·17-s + 1.14·19-s − 0.426·22-s − 0.834·23-s + 0.196·26-s + 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.857·34-s + 0.657·37-s + 0.811·38-s + 0.468·41-s + 1.67·43-s − 0.301·44-s − 0.589·46-s + 0.291·47-s + 0.138·52-s − 0.824·53-s + 0.262·58-s + 0.781·59-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.384926251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.384926251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−12.67917813932837, −12.43582046243695, −11.82746747041095, −11.34172966018200, −11.00693333208005, −10.62888577606515, −9.910686626583208, −9.630931870993208, −9.089434731768212, −8.426972184690198, −7.995887465229388, −7.632534040468041, −6.989647531061121, −6.546589940136528, −6.109550543311592, −5.515936960223627, −5.155290801940360, −4.557033622062694, −4.024981177447809, −3.702249931590108, −2.735099587140164, −2.628910648735625, −1.948995965877139, −1.131184642810546, −0.5206274634854719,
0.5206274634854719, 1.131184642810546, 1.948995965877139, 2.628910648735625, 2.735099587140164, 3.702249931590108, 4.024981177447809, 4.557033622062694, 5.155290801940360, 5.515936960223627, 6.109550543311592, 6.546589940136528, 6.989647531061121, 7.632534040468041, 7.995887465229388, 8.426972184690198, 9.089434731768212, 9.630931870993208, 9.910686626583208, 10.62888577606515, 11.00693333208005, 11.34172966018200, 11.82746747041095, 12.43582046243695, 12.67917813932837
