L(s) = 1 | − 2.61i·3-s + 1.08·5-s + (−1.08 + 2.41i)7-s − 3.82·9-s − 2·11-s + 4.14·13-s − 2.82i·15-s − 7.39i·17-s + 4.77i·19-s + (6.30 + 2.82i)21-s − 3.65i·23-s − 3.82·25-s + 2.16i·27-s − 7.65i·29-s − 7.39·31-s + ⋯ |
L(s) = 1 | − 1.50i·3-s + 0.484·5-s + (−0.409 + 0.912i)7-s − 1.27·9-s − 0.603·11-s + 1.14·13-s − 0.730i·15-s − 1.79i·17-s + 1.09i·19-s + (1.37 + 0.617i)21-s − 0.762i·23-s − 0.765·25-s + 0.416i·27-s − 1.42i·29-s − 1.32·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.235092534\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.235092534\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.08 - 2.41i)T \) |
good | 3 | \( 1 + 2.61iT - 3T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.14T + 13T^{2} \) |
| 17 | \( 1 + 7.39iT - 17T^{2} \) |
| 19 | \( 1 - 4.77iT - 19T^{2} \) |
| 23 | \( 1 + 3.65iT - 23T^{2} \) |
| 29 | \( 1 + 7.65iT - 29T^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 + 3.65iT - 37T^{2} \) |
| 41 | \( 1 + 8.28iT - 41T^{2} \) |
| 43 | \( 1 + 7.65T + 43T^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 5.67iT - 59T^{2} \) |
| 61 | \( 1 - 1.08T + 61T^{2} \) |
| 67 | \( 1 - 4.34T + 67T^{2} \) |
| 71 | \( 1 + 3.17iT - 71T^{2} \) |
| 73 | \( 1 - 0.896iT - 73T^{2} \) |
| 79 | \( 1 + 7.17iT - 79T^{2} \) |
| 83 | \( 1 - 1.71iT - 83T^{2} \) |
| 89 | \( 1 - 5.22iT - 89T^{2} \) |
| 97 | \( 1 - 11.7iT - 97T^{2} \) |
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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
−8.825811484481268499924550633366, −8.022375215931806682344923173202, −7.38174852702836586365188522619, −6.44998049252449570973447315922, −5.90651206697771614257240258370, −5.24037410199597891114769068911, −3.63755811918544478688169572852, −2.47779054111912816447430849280, −1.90138917277985099365857797382, −0.44343076308839292133651205782,
1.56504977429493433737905216890, 3.24965842138746688402565418292, 3.73143451930707084041597305558, 4.61005800461538436608208020228, 5.50183325022583731886636862305, 6.23679250201040934522524430530, 7.23037770340926917820615492842, 8.340524517532065057497916972459, 8.963010656437935881783080956534, 9.830498490764541792348679092253