L(s) = 1 | + (1.24 − 2.15i)5-s + (0.909 + 1.57i)7-s + (−0.598 − 1.03i)11-s + (−2.83 + 4.90i)13-s + 5.30·17-s − 4.55·19-s + (2.01 − 3.48i)23-s + (−0.589 − 1.02i)25-s + (3.01 + 5.22i)29-s + (2.81 − 4.87i)31-s + 4.51·35-s + 5.18·37-s + (−4.57 + 7.92i)41-s + (3.99 + 6.91i)43-s + (−1.39 − 2.41i)47-s + ⋯ |
L(s) = 1 | + (0.555 − 0.962i)5-s + (0.343 + 0.595i)7-s + (−0.180 − 0.312i)11-s + (−0.785 + 1.36i)13-s + 1.28·17-s − 1.04·19-s + (0.419 − 0.727i)23-s + (−0.117 − 0.204i)25-s + (0.559 + 0.969i)29-s + (0.505 − 0.876i)31-s + 0.763·35-s + 0.851·37-s + (−0.714 + 1.23i)41-s + (0.608 + 1.05i)43-s + (−0.203 − 0.352i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.110880757\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.110880757\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.24 + 2.15i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.909 - 1.57i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.598 + 1.03i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.83 - 4.90i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 + 4.55T + 19T^{2} \) |
| 23 | \( 1 + (-2.01 + 3.48i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.01 - 5.22i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.81 + 4.87i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.18T + 37T^{2} \) |
| 41 | \( 1 + (4.57 - 7.92i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.99 - 6.91i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.39 + 2.41i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 1.54T + 53T^{2} \) |
| 59 | \( 1 + (-1.85 + 3.21i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.01 - 6.95i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.91 - 11.9i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + (-4.36 - 7.56i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.89 + 15.4i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 0.455T + 89T^{2} \) |
| 97 | \( 1 + (1.01 + 1.76i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.535672527402998949541727512986, −8.183146710083881340101993509748, −7.07292678820658866070702148274, −6.30649689990894419125300308378, −5.50503078720474773269819121605, −4.84204514130089566105808499401, −4.22427641009708728094898985301, −2.85965023552023820535804307541, −2.01149016307354946715447224776, −1.01564498948338351786456906851,
0.76304439616245481127212434305, 2.15147069885598079822527056250, 2.91145824604627158898758039916, 3.76318604652770139689227199174, 4.86480446578587819888501532553, 5.54641156732574473736219495632, 6.36471538874023163174469229957, 7.15355262330961657803891739224, 7.74601828413673877529080592840, 8.349320275570190197179622108161