L(s) = 1 | + (−1.32 + 0.5i)2-s + (1.50 − 1.32i)4-s − i·5-s + 2·7-s + (−1.32 + 2.50i)8-s + (0.5 + 1.32i)10-s − 2i·11-s + (−2.64 + i)14-s + (0.500 − 3.96i)16-s − 5.29i·19-s + (−1.32 − 1.50i)20-s + (1 + 2.64i)22-s + 5.29·23-s − 25-s + (3.00 − 2.64i)28-s − 6i·29-s + ⋯ |
L(s) = 1 | + (−0.935 + 0.353i)2-s + (0.750 − 0.661i)4-s − 0.447i·5-s + 0.755·7-s + (−0.467 + 0.883i)8-s + (0.158 + 0.418i)10-s − 0.603i·11-s + (−0.707 + 0.267i)14-s + (0.125 − 0.992i)16-s − 1.21i·19-s + (−0.295 − 0.335i)20-s + (0.213 + 0.564i)22-s + 1.10·23-s − 0.200·25-s + (0.566 − 0.499i)28-s − 1.11i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.918322 - 0.227989i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.918322 - 0.227989i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.32 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.29iT - 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10.5iT - 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 10.5iT - 43T^{2} \) |
| 47 | \( 1 - 5.29T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 10iT - 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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
−11.27132196395851703726953625619, −10.41337216527818415061976883347, −9.294464979345681713024557015371, −8.609013423775134402833651769980, −7.79978927213786820934880663446, −6.77137566652881648478023732742, −5.63699143439207591158254968529, −4.61668749150245551957992467265, −2.64320459199366440824974384142, −0.994883475537477623186803082558,
1.54740601209356671730314881445, 2.91127859826938712957343690749, 4.32428788257918239087071333258, 5.87982453619531752534086852503, 7.12877778983269025059666269001, 7.78012344970630886962265511538, 8.801656035346736390745595073924, 9.712828963180114982055977011016, 10.65765938299176352109103215428, 11.22199967987096879872741092170