| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s − 2·9-s − 10-s − 12-s + 2.46·13-s + 14-s − 15-s + 16-s + 0.535·17-s + 2·18-s + 7.92·19-s + 20-s + 21-s + 2.46·23-s + 24-s + 25-s − 2.46·26-s + 5·27-s − 28-s − 6.92·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 0.666·9-s − 0.316·10-s − 0.288·12-s + 0.683·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s + 0.129·17-s + 0.471·18-s + 1.81·19-s + 0.223·20-s + 0.218·21-s + 0.513·23-s + 0.204·24-s + 0.200·25-s − 0.483·26-s + 0.962·27-s − 0.188·28-s − 1.28·29-s + 0.182·30-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 - 0.535T + 17T^{2} \) |
| 19 | \( 1 - 7.92T + 19T^{2} \) |
| 23 | \( 1 - 2.46T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 7.46T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 + 8.39T + 43T^{2} \) |
| 47 | \( 1 + 7.46T + 47T^{2} \) |
| 53 | \( 1 + 0.535T + 53T^{2} \) |
| 59 | \( 1 + 3.92T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.40822335760396283526976106681, −6.76480522608684036184089124470, −6.11073024375040691970898673995, −5.44092763489852699412642181937, −5.02381906710377050033545206810, −3.50238681027965812610083168346, −3.18585464916773625430629668266, −1.97848347504732522514644293968, −1.10804353101182654681588083505, 0,
1.10804353101182654681588083505, 1.97848347504732522514644293968, 3.18585464916773625430629668266, 3.50238681027965812610083168346, 5.02381906710377050033545206810, 5.44092763489852699412642181937, 6.11073024375040691970898673995, 6.76480522608684036184089124470, 7.40822335760396283526976106681
