| L(s) = 1 | + 1.80·2-s + 0.896·3-s + 1.26·4-s + 3.66·5-s + 1.62·6-s + 2.54·7-s − 1.32·8-s − 2.19·9-s + 6.61·10-s − 3.90·11-s + 1.13·12-s + 13-s + 4.60·14-s + 3.28·15-s − 4.92·16-s + 2.48·17-s − 3.97·18-s + 7.78·19-s + 4.64·20-s + 2.28·21-s − 7.06·22-s + 3.72·23-s − 1.18·24-s + 8.40·25-s + 1.80·26-s − 4.65·27-s + 3.22·28-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 0.517·3-s + 0.633·4-s + 1.63·5-s + 0.661·6-s + 0.962·7-s − 0.468·8-s − 0.732·9-s + 2.09·10-s − 1.17·11-s + 0.327·12-s + 0.277·13-s + 1.22·14-s + 0.847·15-s − 1.23·16-s + 0.602·17-s − 0.935·18-s + 1.78·19-s + 1.03·20-s + 0.497·21-s − 1.50·22-s + 0.776·23-s − 0.242·24-s + 1.68·25-s + 0.354·26-s − 0.896·27-s + 0.609·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.761610301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.761610301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 3 | \( 1 - 0.896T + 3T^{2} \) |
| 5 | \( 1 - 3.66T + 5T^{2} \) |
| 7 | \( 1 - 2.54T + 7T^{2} \) |
| 11 | \( 1 + 3.90T + 11T^{2} \) |
| 17 | \( 1 - 2.48T + 17T^{2} \) |
| 19 | \( 1 - 7.78T + 19T^{2} \) |
| 23 | \( 1 - 3.72T + 23T^{2} \) |
| 29 | \( 1 + 0.745T + 29T^{2} \) |
| 31 | \( 1 - 8.09T + 31T^{2} \) |
| 37 | \( 1 + 4.72T + 37T^{2} \) |
| 41 | \( 1 + 7.63T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 7.98T + 47T^{2} \) |
| 53 | \( 1 - 6.95T + 53T^{2} \) |
| 59 | \( 1 - 8.39T + 59T^{2} \) |
| 61 | \( 1 + 8.67T + 61T^{2} \) |
| 67 | \( 1 + 2.94T + 67T^{2} \) |
| 71 | \( 1 - 5.18T + 71T^{2} \) |
| 73 | \( 1 + 8.69T + 73T^{2} \) |
| 79 | \( 1 + 9.61T + 79T^{2} \) |
| 83 | \( 1 + 9.31T + 83T^{2} \) |
| 89 | \( 1 - 0.0871T + 89T^{2} \) |
| 97 | \( 1 - 0.899T + 97T^{2} \) |
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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
−9.705619365225613012747962360567, −8.749803932267261825703791184172, −8.101652985450469777454682815854, −6.92309961772874805240192142820, −5.86502281272843520320860234130, −5.24441205845509554812875919097, −4.95472579548394469564044825398, −3.25352658725966164587144200279, −2.76424156603105663567592388316, −1.61575401193393429216191027446,
1.61575401193393429216191027446, 2.76424156603105663567592388316, 3.25352658725966164587144200279, 4.95472579548394469564044825398, 5.24441205845509554812875919097, 5.86502281272843520320860234130, 6.92309961772874805240192142820, 8.101652985450469777454682815854, 8.749803932267261825703791184172, 9.705619365225613012747962360567
