| L(s) = 1 | + 1.76·2-s − 3-s + 1.12·4-s − 2.50·5-s − 1.76·6-s − 4.08·7-s − 1.55·8-s + 9-s − 4.42·10-s − 11-s − 1.12·12-s + 4.58·13-s − 7.22·14-s + 2.50·15-s − 4.98·16-s − 1.68·17-s + 1.76·18-s + 2.54·19-s − 2.80·20-s + 4.08·21-s − 1.76·22-s + 6.64·23-s + 1.55·24-s + 1.26·25-s + 8.09·26-s − 27-s − 4.58·28-s + ⋯ |
| L(s) = 1 | + 1.24·2-s − 0.577·3-s + 0.560·4-s − 1.11·5-s − 0.721·6-s − 1.54·7-s − 0.549·8-s + 0.333·9-s − 1.39·10-s − 0.301·11-s − 0.323·12-s + 1.27·13-s − 1.93·14-s + 0.646·15-s − 1.24·16-s − 0.409·17-s + 0.416·18-s + 0.583·19-s − 0.626·20-s + 0.892·21-s − 0.376·22-s + 1.38·23-s + 0.317·24-s + 0.252·25-s + 1.58·26-s − 0.192·27-s − 0.865·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.422589918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.422589918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.76T + 2T^{2} \) |
| 5 | \( 1 + 2.50T + 5T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 + 1.68T + 17T^{2} \) |
| 19 | \( 1 - 2.54T + 19T^{2} \) |
| 23 | \( 1 - 6.64T + 23T^{2} \) |
| 29 | \( 1 - 2.87T + 29T^{2} \) |
| 31 | \( 1 - 5.77T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 7.50T + 41T^{2} \) |
| 43 | \( 1 + 5.62T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 + 6.73T + 53T^{2} \) |
| 59 | \( 1 - 1.44T + 59T^{2} \) |
| 67 | \( 1 + 2.89T + 67T^{2} \) |
| 71 | \( 1 + 3.64T + 71T^{2} \) |
| 73 | \( 1 + 1.53T + 73T^{2} \) |
| 79 | \( 1 - 6.59T + 79T^{2} \) |
| 83 | \( 1 - 4.68T + 83T^{2} \) |
| 89 | \( 1 - 8.01T + 89T^{2} \) |
| 97 | \( 1 + 8.48T + 97T^{2} \) |
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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
−9.153333743595949078783839586557, −8.348276552000010962303535691269, −7.23791322016334368645353769321, −6.47715261463643550116153960991, −6.02629059270414192783291405960, −4.99740373848448654919878432942, −4.21863459331001274464655987182, −3.44151257025713308139993053821, −2.89866242137100939648060531828, −0.65909303968584254236637041802,
0.65909303968584254236637041802, 2.89866242137100939648060531828, 3.44151257025713308139993053821, 4.21863459331001274464655987182, 4.99740373848448654919878432942, 6.02629059270414192783291405960, 6.47715261463643550116153960991, 7.23791322016334368645353769321, 8.348276552000010962303535691269, 9.153333743595949078783839586557
