L(s) = 1 | + (−2.05 − 1.18i)5-s + (−4.05 − 7.02i)7-s + (17.6 − 10.1i)11-s + (3.05 − 5.29i)13-s + 17.9i·17-s + 9.11·19-s + (29.0 + 16.7i)23-s + (−9.67 − 16.7i)25-s + (14.4 − 8.31i)29-s + (−11.1 + 19.3i)31-s + 19.2i·35-s + 50.4·37-s + (−29.9 − 17.3i)41-s + (−11.5 − 19.9i)43-s + (−33.1 + 19.1i)47-s + ⋯ |
L(s) = 1 | + (−0.411 − 0.237i)5-s + (−0.579 − 1.00i)7-s + (1.60 − 0.924i)11-s + (0.235 − 0.407i)13-s + 1.05i·17-s + 0.479·19-s + (1.26 + 0.729i)23-s + (−0.387 − 0.670i)25-s + (0.496 − 0.286i)29-s + (−0.360 + 0.624i)31-s + 0.551i·35-s + 1.36·37-s + (−0.730 − 0.421i)41-s + (−0.267 − 0.463i)43-s + (−0.705 + 0.407i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.118 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.772295750\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.772295750\) |
\(L(2)\) |
|
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 | \( 1 + (2.05 + 1.18i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (4.05 + 7.02i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-17.6 + 10.1i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-3.05 + 5.29i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 17.9iT - 289T^{2} \) |
| 19 | \( 1 - 9.11T + 361T^{2} \) |
| 23 | \( 1 + (-29.0 - 16.7i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-14.4 + 8.31i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (11.1 - 19.3i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 50.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (29.9 + 17.3i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (11.5 + 19.9i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (33.1 - 19.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 19.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-2.96 - 1.71i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (23.1 + 40.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-3.14 + 5.45i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 35.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 47.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + (42.2 + 73.2i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-33.1 + 19.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 143. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (40.3 + 69.9i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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
−8.865767162962206568065290496028, −8.217007042993824738067693950991, −7.28095592814327677863778528273, −6.52646043237131888733504415596, −5.86678053727415801639265798965, −4.61062014964439474678401348619, −3.70672378996668540474195173020, −3.28781930031640066246578010008, −1.41936468412941203688483587870, −0.56191598081623322359681382188,
1.16592441018515295145176164184, 2.46600748760413630053698014627, 3.37577529330095859579972612826, 4.35727904898428918456104389303, 5.22842794149304591498578158226, 6.37820347387952298095029936244, 6.82248087128381082768735260142, 7.66486028240000518856111496832, 8.808861169727431479469490563859, 9.358741712538943927085379339831