| L(s) = 1 | − 0.546·2-s − 3-s − 1.70·4-s − 0.842·5-s + 0.546·6-s − 4.19·7-s + 2.02·8-s + 9-s + 0.460·10-s − 11-s + 1.70·12-s − 5.95·13-s + 2.29·14-s + 0.842·15-s + 2.29·16-s + 0.386·17-s − 0.546·18-s − 2.95·19-s + 1.43·20-s + 4.19·21-s + 0.546·22-s − 0.974·23-s − 2.02·24-s − 4.28·25-s + 3.25·26-s − 27-s + 7.13·28-s + ⋯ |
| L(s) = 1 | − 0.386·2-s − 0.577·3-s − 0.850·4-s − 0.376·5-s + 0.223·6-s − 1.58·7-s + 0.714·8-s + 0.333·9-s + 0.145·10-s − 0.301·11-s + 0.491·12-s − 1.65·13-s + 0.612·14-s + 0.217·15-s + 0.574·16-s + 0.0936·17-s − 0.128·18-s − 0.677·19-s + 0.320·20-s + 0.914·21-s + 0.116·22-s − 0.203·23-s − 0.412·24-s − 0.857·25-s + 0.638·26-s − 0.192·27-s + 1.34·28-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
| \(L(1)\) |
\(\approx\) |
\(0.05230048157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05230048157\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.546T + 2T^{2} \) |
| 5 | \( 1 + 0.842T + 5T^{2} \) |
| 7 | \( 1 + 4.19T + 7T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 17 | \( 1 - 0.386T + 17T^{2} \) |
| 19 | \( 1 + 2.95T + 19T^{2} \) |
| 23 | \( 1 + 0.974T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + 3.16T + 37T^{2} \) |
| 41 | \( 1 + 1.29T + 41T^{2} \) |
| 43 | \( 1 + 4.99T + 43T^{2} \) |
| 47 | \( 1 - 7.87T + 47T^{2} \) |
| 53 | \( 1 - 0.370T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 67 | \( 1 - 9.80T + 67T^{2} \) |
| 71 | \( 1 + 4.36T + 71T^{2} \) |
| 73 | \( 1 + 2.06T + 73T^{2} \) |
| 79 | \( 1 + 8.67T + 79T^{2} \) |
| 83 | \( 1 + 7.09T + 83T^{2} \) |
| 89 | \( 1 + 7.42T + 89T^{2} \) |
| 97 | \( 1 + 6.00T + 97T^{2} \) |
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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
−9.382187269403932530070625960506, −8.492517768723624612275295232303, −7.33997076190643201404933325176, −7.13564382958697804397076922277, −5.80719643349655716480389434354, −5.28122190819204506430244460149, −4.14550132509359554729056607821, −3.52061976786495961496082917082, −2.11953712598252897469492866384, −0.15608624392329789066675289308,
0.15608624392329789066675289308, 2.11953712598252897469492866384, 3.52061976786495961496082917082, 4.14550132509359554729056607821, 5.28122190819204506430244460149, 5.80719643349655716480389434354, 7.13564382958697804397076922277, 7.33997076190643201404933325176, 8.492517768723624612275295232303, 9.382187269403932530070625960506
