| L(s) = 1 | + (1.99 + 0.150i)2-s + (−1.64 − 2.50i)3-s + (3.95 + 0.600i)4-s + (0.451 − 4.97i)5-s + (−2.89 − 5.25i)6-s + (2.33 − 2.33i)7-s + (7.79 + 1.79i)8-s + (−3.59 + 8.24i)9-s + (1.64 − 9.86i)10-s + (2.36 − 7.27i)11-s + (−4.99 − 10.9i)12-s + (−21.0 − 10.7i)13-s + (5.00 − 4.29i)14-s + (−13.2 + 7.05i)15-s + (15.2 + 4.75i)16-s + (−3.31 + 20.9i)17-s + ⋯ |
| L(s) = 1 | + (0.997 + 0.0752i)2-s + (−0.547 − 0.836i)3-s + (0.988 + 0.150i)4-s + (0.0902 − 0.995i)5-s + (−0.483 − 0.875i)6-s + (0.333 − 0.333i)7-s + (0.974 + 0.224i)8-s + (−0.399 + 0.916i)9-s + (0.164 − 0.986i)10-s + (0.214 − 0.661i)11-s + (−0.415 − 0.909i)12-s + (−1.61 − 0.824i)13-s + (0.357 − 0.307i)14-s + (−0.882 + 0.470i)15-s + (0.954 + 0.296i)16-s + (−0.195 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.64968 - 1.87695i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64968 - 1.87695i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.99 - 0.150i)T \) |
| 3 | \( 1 + (1.64 + 2.50i)T \) |
| 5 | \( 1 + (-0.451 + 4.97i)T \) |
| good | 7 | \( 1 + (-2.33 + 2.33i)T - 49iT^{2} \) |
| 11 | \( 1 + (-2.36 + 7.27i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (21.0 + 10.7i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (3.31 - 20.9i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-25.5 + 18.5i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-3.56 - 6.99i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (2.06 + 1.50i)T + (259. + 799. i)T^{2} \) |
| 31 | \( 1 + (33.1 + 45.6i)T + (-296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-7.84 + 15.3i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-64.0 + 20.8i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-31.2 - 31.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-10.1 - 64.1i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-8.86 - 55.9i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-22.8 + 7.41i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (0.860 - 2.64i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (1.58 - 10.0i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-12.6 - 9.17i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-25.7 - 50.4i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-69.9 - 50.8i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (2.15 - 13.5i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-27.5 + 84.6i)T + (-6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (4.83 + 30.5i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
| 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
−11.48732686516481135921463763221, −10.83066357050089408830391953976, −9.390452578653648005049482459212, −7.82286264963755518862353046052, −7.46625102875725274929144636446, −5.95348660076214295977782988548, −5.34928846507249303182029326038, −4.29719722650811616177113287856, −2.51242931385455795780700811902, −0.977647601333025687223631907859,
2.30362806248983812381091308047, 3.52835509811618799461353190153, 4.78206612275047601555339984989, 5.46886308827557542039677557267, 6.80720700622501719140343204936, 7.38439708174487746309206424296, 9.427461180130431126247458368323, 10.05495224881054196549093488461, 11.04556538295881093544607116512, 11.90897331210021152813301905245
