| L(s) = 1 | + 2.26·2-s + 3-s + 3.14·4-s − 2.85·5-s + 2.26·6-s + 7-s + 2.60·8-s + 9-s − 6.48·10-s − 4.41·11-s + 3.14·12-s + 5.05·13-s + 2.26·14-s − 2.85·15-s − 0.391·16-s − 4.39·17-s + 2.26·18-s − 2.41·19-s − 8.99·20-s + 21-s − 10.0·22-s − 0.472·23-s + 2.60·24-s + 3.16·25-s + 11.4·26-s + 27-s + 3.14·28-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 0.577·3-s + 1.57·4-s − 1.27·5-s + 0.926·6-s + 0.377·7-s + 0.919·8-s + 0.333·9-s − 2.05·10-s − 1.33·11-s + 0.908·12-s + 1.40·13-s + 0.606·14-s − 0.737·15-s − 0.0978·16-s − 1.06·17-s + 0.534·18-s − 0.553·19-s − 2.01·20-s + 0.218·21-s − 2.13·22-s − 0.0984·23-s + 0.531·24-s + 0.633·25-s + 2.24·26-s + 0.192·27-s + 0.594·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 2.26T + 2T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 11 | \( 1 + 4.41T + 11T^{2} \) |
| 13 | \( 1 - 5.05T + 13T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 + 0.472T + 23T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 - 1.50T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 + 3.53T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 1.39T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 15.0T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 7.92T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−7.53327547114178525887458568531, −6.60080027270589706483239584576, −6.05730946063612679662956122589, −5.11585763918403042437911145018, −4.44566592970927690629846103280, −4.05979535213669690534990728467, −3.24965318567314546546827154845, −2.69970631709127327931089813725, −1.68778123134091348931438754323, 0,
1.68778123134091348931438754323, 2.69970631709127327931089813725, 3.24965318567314546546827154845, 4.05979535213669690534990728467, 4.44566592970927690629846103280, 5.11585763918403042437911145018, 6.05730946063612679662956122589, 6.60080027270589706483239584576, 7.53327547114178525887458568531
