L(s) = 1 | − 2.64·2-s − 0.354·3-s + 4.97·4-s − 1.23·5-s + 0.934·6-s − 3.87·7-s − 7.84·8-s − 2.87·9-s + 3.25·10-s + 2.01·11-s − 1.75·12-s − 3.33·13-s + 10.2·14-s + 0.436·15-s + 10.7·16-s − 1.76·17-s + 7.58·18-s − 2.88·19-s − 6.12·20-s + 1.37·21-s − 5.31·22-s − 7.07·23-s + 2.77·24-s − 3.48·25-s + 8.79·26-s + 2.07·27-s − 19.2·28-s + ⋯ |
L(s) = 1 | − 1.86·2-s − 0.204·3-s + 2.48·4-s − 0.551·5-s + 0.381·6-s − 1.46·7-s − 2.77·8-s − 0.958·9-s + 1.02·10-s + 0.606·11-s − 0.507·12-s − 0.923·13-s + 2.73·14-s + 0.112·15-s + 2.69·16-s − 0.428·17-s + 1.78·18-s − 0.661·19-s − 1.36·20-s + 0.299·21-s − 1.13·22-s − 1.47·23-s + 0.566·24-s − 0.696·25-s + 1.72·26-s + 0.400·27-s − 3.63·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.06932766464\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06932766464\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 3 | \( 1 + 0.354T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 11 | \( 1 - 2.01T + 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 17 | \( 1 + 1.76T + 17T^{2} \) |
| 19 | \( 1 + 2.88T + 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 - 1.16T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 - 0.396T + 37T^{2} \) |
| 41 | \( 1 + 1.94T + 41T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 - 5.58T + 53T^{2} \) |
| 59 | \( 1 - 6.47T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 5.78T + 67T^{2} \) |
| 71 | \( 1 - 3.37T + 71T^{2} \) |
| 73 | \( 1 + 2.47T + 73T^{2} \) |
| 79 | \( 1 - 8.62T + 79T^{2} \) |
| 83 | \( 1 - 3.06T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−9.312500859197404866913369954800, −8.568801540775188414384241090013, −7.86059154626172799685217044947, −7.02209448650297564688992751515, −6.41193321286963819684824612953, −5.71867025218730168178026295887, −3.97395045936445593474023328203, −2.95347816062646027013402163227, −2.01474731023772145762479867229, −0.21885272567384162058840849749,
0.21885272567384162058840849749, 2.01474731023772145762479867229, 2.95347816062646027013402163227, 3.97395045936445593474023328203, 5.71867025218730168178026295887, 6.41193321286963819684824612953, 7.02209448650297564688992751515, 7.86059154626172799685217044947, 8.568801540775188414384241090013, 9.312500859197404866913369954800