| L(s) = 1 | − 1.36·2-s − 3-s − 0.141·4-s + 1.36·6-s − 2.50·7-s + 2.91·8-s + 9-s + 11-s + 0.141·12-s − 1.14·13-s + 3.41·14-s − 3.69·16-s − 7.64·17-s − 1.36·18-s + 1.77·19-s + 2.50·21-s − 1.36·22-s − 1.41·23-s − 2.91·24-s + 1.55·26-s − 27-s + 0.353·28-s − 0.726·29-s + 2.85·31-s − 0.797·32-s − 33-s + 10.4·34-s + ⋯ |
| L(s) = 1 | − 0.964·2-s − 0.577·3-s − 0.0706·4-s + 0.556·6-s − 0.946·7-s + 1.03·8-s + 0.333·9-s + 0.301·11-s + 0.0408·12-s − 0.316·13-s + 0.912·14-s − 0.924·16-s − 1.85·17-s − 0.321·18-s + 0.407·19-s + 0.546·21-s − 0.290·22-s − 0.294·23-s − 0.595·24-s + 0.305·26-s − 0.192·27-s + 0.0668·28-s − 0.134·29-s + 0.513·31-s − 0.141·32-s − 0.174·33-s + 1.78·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4719216499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4719216499\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 13 | \( 1 + 1.14T + 13T^{2} \) |
| 17 | \( 1 + 7.64T + 17T^{2} \) |
| 19 | \( 1 - 1.77T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 0.726T + 29T^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 - 8.42T + 37T^{2} \) |
| 41 | \( 1 - 0.636T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 + 6.14T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 3.41T + 59T^{2} \) |
| 61 | \( 1 - 4.59T + 61T^{2} \) |
| 67 | \( 1 - 9.32T + 67T^{2} \) |
| 71 | \( 1 - 5.85T + 71T^{2} \) |
| 73 | \( 1 + 7.55T + 73T^{2} \) |
| 79 | \( 1 - 6.91T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 - 3.45T + 89T^{2} \) |
| 97 | \( 1 - 19.4T + 97T^{2} \) |
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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
−10.03034025245351059706082093845, −9.426984253935756024914684440040, −8.775098137802566346648366262963, −7.70204594248208799903047377716, −6.84558157470127505151102660516, −6.09479694912721430877460294668, −4.80672537665227115967301278157, −3.95701434672746778984827673068, −2.31824531880050942501510202791, −0.65095342717078505375832497646,
0.65095342717078505375832497646, 2.31824531880050942501510202791, 3.95701434672746778984827673068, 4.80672537665227115967301278157, 6.09479694912721430877460294668, 6.84558157470127505151102660516, 7.70204594248208799903047377716, 8.775098137802566346648366262963, 9.426984253935756024914684440040, 10.03034025245351059706082093845
