L(s) = 1 | + 4.93·7-s + 2.41·11-s + 2.90·13-s − 6.86·17-s − 4.17·19-s − 3.35·23-s − 5.19·29-s − 6.17·31-s + 7.84·37-s + 5.87·41-s + 4.93·43-s + 11.9·47-s + 17.3·49-s − 8.54·53-s + 1.05·59-s + 9.17·61-s + 4.05·67-s + 14.1·71-s + 2.02·73-s + 11.9·77-s + 6·79-s + 5.18·83-s − 3.09·89-s + 14.3·91-s + 0.882·97-s − 5.87·101-s + 9.87·103-s + ⋯ |
L(s) = 1 | + 1.86·7-s + 0.727·11-s + 0.806·13-s − 1.66·17-s − 0.958·19-s − 0.700·23-s − 0.964·29-s − 1.10·31-s + 1.28·37-s + 0.917·41-s + 0.752·43-s + 1.73·47-s + 2.47·49-s − 1.17·53-s + 0.136·59-s + 1.17·61-s + 0.495·67-s + 1.68·71-s + 0.237·73-s + 1.35·77-s + 0.675·79-s + 0.569·83-s − 0.327·89-s + 1.50·91-s + 0.0896·97-s − 0.584·101-s + 0.972·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.705192075\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.705192075\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 4.93T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 + 6.86T + 17T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 + 3.35T + 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 + 6.17T + 31T^{2} \) |
| 37 | \( 1 - 7.84T + 37T^{2} \) |
| 41 | \( 1 - 5.87T + 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 8.54T + 53T^{2} \) |
| 59 | \( 1 - 1.05T + 59T^{2} \) |
| 61 | \( 1 - 9.17T + 61T^{2} \) |
| 67 | \( 1 - 4.05T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 2.02T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 5.18T + 83T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 - 0.882T + 97T^{2} \) |
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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
−7.85639294059716394175211444464, −7.25498983280086564482978598173, −6.36374372228394402820003421583, −5.79953832096996883790006794303, −4.93215888408953833512314096305, −4.14166353399714198403695902569, −3.93001536240127252580404912333, −2.26339060379925072646719808992, −1.93462420000399099692621897370, −0.835815594156000659035601418363,
0.835815594156000659035601418363, 1.93462420000399099692621897370, 2.26339060379925072646719808992, 3.93001536240127252580404912333, 4.14166353399714198403695902569, 4.93215888408953833512314096305, 5.79953832096996883790006794303, 6.36374372228394402820003421583, 7.25498983280086564482978598173, 7.85639294059716394175211444464