L(s) = 1 | − 5-s + 3·7-s − 4.87·11-s − 2.87·13-s − 3.87·17-s − 2.87·19-s + 23-s + 25-s + 5.87·29-s + 3·31-s − 3·35-s + 37-s + 5.87·41-s + 3.74·43-s + 6.87·47-s + 2·49-s + 3.87·53-s + 4.87·55-s − 5.87·59-s + 8.87·61-s + 2.87·65-s + 10.7·67-s + 13.6·71-s − 2.87·73-s − 14.6·77-s + 4·79-s + 0.127·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.13·7-s − 1.46·11-s − 0.796·13-s − 0.939·17-s − 0.659·19-s + 0.208·23-s + 0.200·25-s + 1.09·29-s + 0.538·31-s − 0.507·35-s + 0.164·37-s + 0.917·41-s + 0.571·43-s + 1.00·47-s + 0.285·49-s + 0.531·53-s + 0.657·55-s − 0.764·59-s + 1.13·61-s + 0.356·65-s + 1.31·67-s + 1.61·71-s − 0.336·73-s − 1.66·77-s + 0.450·79-s + 0.0139·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.499687424\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.499687424\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 29 | \( 1 - 5.87T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 5.87T + 41T^{2} \) |
| 43 | \( 1 - 3.74T + 43T^{2} \) |
| 47 | \( 1 - 6.87T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 + 5.87T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 2.87T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 0.127T + 83T^{2} \) |
| 89 | \( 1 + 1.74T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.286186221497427599109269680358, −7.78566296946842476152463429358, −7.14066174885916695706163759629, −6.23226735236534657660091952639, −5.17549083536370361306094858279, −4.77166249015211022164479138542, −4.02708314585323576082983517451, −2.67342477661446101152202137561, −2.19320259583855989671831305463, −0.67597643643831452414558733457,
0.67597643643831452414558733457, 2.19320259583855989671831305463, 2.67342477661446101152202137561, 4.02708314585323576082983517451, 4.77166249015211022164479138542, 5.17549083536370361306094858279, 6.23226735236534657660091952639, 7.14066174885916695706163759629, 7.78566296946842476152463429358, 8.286186221497427599109269680358