| L(s) = 1 | − 2-s − 1.96·3-s + 4-s − 1.21·5-s + 1.96·6-s − 3.03·7-s − 8-s + 0.865·9-s + 1.21·10-s − 3.99·11-s − 1.96·12-s − 2.32·13-s + 3.03·14-s + 2.38·15-s + 16-s − 1.92·17-s − 0.865·18-s + 0.125·19-s − 1.21·20-s + 5.96·21-s + 3.99·22-s + 8.03·23-s + 1.96·24-s − 3.53·25-s + 2.32·26-s + 4.19·27-s − 3.03·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.13·3-s + 0.5·4-s − 0.542·5-s + 0.802·6-s − 1.14·7-s − 0.353·8-s + 0.288·9-s + 0.383·10-s − 1.20·11-s − 0.567·12-s − 0.644·13-s + 0.811·14-s + 0.615·15-s + 0.250·16-s − 0.467·17-s − 0.204·18-s + 0.0287·19-s − 0.271·20-s + 1.30·21-s + 0.851·22-s + 1.67·23-s + 0.401·24-s − 0.706·25-s + 0.455·26-s + 0.807·27-s − 0.573·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 1.96T + 3T^{2} \) |
| 5 | \( 1 + 1.21T + 5T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 11 | \( 1 + 3.99T + 11T^{2} \) |
| 13 | \( 1 + 2.32T + 13T^{2} \) |
| 17 | \( 1 + 1.92T + 17T^{2} \) |
| 19 | \( 1 - 0.125T + 19T^{2} \) |
| 23 | \( 1 - 8.03T + 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 31 | \( 1 - 5.02T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 - 0.478T + 41T^{2} \) |
| 43 | \( 1 - 6.52T + 43T^{2} \) |
| 47 | \( 1 + 2.89T + 47T^{2} \) |
| 53 | \( 1 + 2.52T + 53T^{2} \) |
| 59 | \( 1 + 7.53T + 59T^{2} \) |
| 61 | \( 1 + 2.67T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 5.48T + 71T^{2} \) |
| 73 | \( 1 - 8.94T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 9.03T + 83T^{2} \) |
| 89 | \( 1 + 1.30T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−8.006300493098167313436745217755, −7.32904334551788364753291402441, −6.56717584659377796268616110299, −6.09400206406843376830251221782, −5.13264694977689231896024536555, −4.51167290245588359600416173169, −3.10648130809290980275948581153, −2.59520158089793925150442067596, −0.821747726617399972938262543845, 0,
0.821747726617399972938262543845, 2.59520158089793925150442067596, 3.10648130809290980275948581153, 4.51167290245588359600416173169, 5.13264694977689231896024536555, 6.09400206406843376830251221782, 6.56717584659377796268616110299, 7.32904334551788364753291402441, 8.006300493098167313436745217755
