L(s) = 1 | + (−1.38 + 0.285i)2-s − 3-s + (1.83 − 0.791i)4-s − 2.03·5-s + (1.38 − 0.285i)6-s + 1.21·7-s + (−2.31 + 1.62i)8-s + 9-s + (2.81 − 0.580i)10-s − 1.88i·11-s + (−1.83 + 0.791i)12-s + 3.19i·13-s + (−1.68 + 0.348i)14-s + 2.03·15-s + (2.74 − 2.90i)16-s − 5.91i·17-s + ⋯ |
L(s) = 1 | + (−0.979 + 0.201i)2-s − 0.577·3-s + (0.918 − 0.395i)4-s − 0.909·5-s + (0.565 − 0.116i)6-s + 0.461·7-s + (−0.819 + 0.572i)8-s + 0.333·9-s + (0.890 − 0.183i)10-s − 0.569i·11-s + (−0.530 + 0.228i)12-s + 0.884i·13-s + (−0.451 + 0.0931i)14-s + 0.524·15-s + (0.687 − 0.726i)16-s − 1.43i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.000288292 - 0.0439175i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.000288292 - 0.0439175i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.38 - 0.285i)T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + (3.96 + 2.69i)T \) |
good | 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 - 1.21T + 7T^{2} \) |
| 11 | \( 1 + 1.88iT - 11T^{2} \) |
| 13 | \( 1 - 3.19iT - 13T^{2} \) |
| 17 | \( 1 + 5.91iT - 17T^{2} \) |
| 19 | \( 1 - 6.08iT - 19T^{2} \) |
| 29 | \( 1 - 9.30iT - 29T^{2} \) |
| 31 | \( 1 + 3.55iT - 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 + 11.6iT - 43T^{2} \) |
| 47 | \( 1 - 11.2iT - 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 0.657T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 2.45iT - 67T^{2} \) |
| 71 | \( 1 - 4.84iT - 71T^{2} \) |
| 73 | \( 1 + 7.87T + 73T^{2} \) |
| 79 | \( 1 - 4.23T + 79T^{2} \) |
| 83 | \( 1 + 7.58iT - 83T^{2} \) |
| 89 | \( 1 - 3.85iT - 89T^{2} \) |
| 97 | \( 1 - 3.03iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.13610105400015078203053893193, −10.45275158217875189071243929380, −9.403317276274144438330261307952, −8.514033923729499835534724172591, −7.68953313948971036507454467884, −6.94690306717380900376075542299, −5.91956535353300518144554760083, −4.81700335978149402298653480108, −3.44382134768630560158368625965, −1.67443756596004994875554732783,
0.03702215470231202576641258872, 1.76459604793647432495545802322, 3.39776665634709373149621777794, 4.57460453243862938937868101250, 5.92328414404867367615334189197, 6.94740695163550803952751381837, 7.88768896070432569498158458824, 8.340930814599248146098661719673, 9.611710868435403265376899977832, 10.42243010895442015847860373573