L(s) = 1 | + (−2.65 + 1.39i)3-s + (−1.41 + 2.45i)7-s + (5.10 − 7.40i)9-s + (−0.949 − 0.547i)11-s + (−2.09 − 3.63i)13-s + 7.85i·17-s + 13.3·19-s + (0.340 − 8.48i)21-s + (21.4 − 12.3i)23-s + (−3.23 + 26.8i)27-s + (−2.43 − 1.40i)29-s + (12.0 + 20.8i)31-s + (3.28 + 0.131i)33-s − 49.9·37-s + (10.6 + 6.72i)39-s + ⋯ |
L(s) = 1 | + (−0.885 + 0.464i)3-s + (−0.202 + 0.350i)7-s + (0.567 − 0.823i)9-s + (−0.0862 − 0.0498i)11-s + (−0.161 − 0.279i)13-s + 0.462i·17-s + 0.703·19-s + (0.0161 − 0.404i)21-s + (0.931 − 0.537i)23-s + (−0.119 + 0.992i)27-s + (−0.0840 − 0.0485i)29-s + (0.389 + 0.673i)31-s + (0.0995 + 0.00398i)33-s − 1.34·37-s + (0.272 + 0.172i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9657345137\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9657345137\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.65 - 1.39i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.41 - 2.45i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (0.949 + 0.547i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (2.09 + 3.63i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 7.85iT - 289T^{2} \) |
| 19 | \( 1 - 13.3T + 361T^{2} \) |
| 23 | \( 1 + (-21.4 + 12.3i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (2.43 + 1.40i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-12.0 - 20.8i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 49.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + (18.9 - 10.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-24.5 + 42.4i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-58.5 - 33.8i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 49.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (86.8 - 50.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (41.5 - 71.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-28.7 - 49.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 14.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 71.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + (54.4 - 94.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-69.7 - 40.2i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 65.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (77.4 - 134. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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
−10.41491517633445380183464663953, −9.355032965857616293910412646084, −8.705515939253145032180028937776, −7.45288379190552957554208186500, −6.62862335531290853869290531944, −5.69186836478314271645265720815, −5.02562625700416431688255713930, −3.95172701407538804931745314106, −2.82788502984070453312656393736, −1.10255380150099953227340490552,
0.41982329922624372693487516231, 1.70999685515513868228049200119, 3.16248565694616857360694761578, 4.49926821822150701467355347371, 5.31236428971822687990959314504, 6.22777685047178678111614036867, 7.17053303804170974022387333661, 7.61865278856266711182308642462, 8.905442220348203441797077425041, 9.792231124920226882981541924314