L(s) = 1 | + 4i·2-s − 16·4-s + (55 + 10i)5-s + 4i·7-s − 64i·8-s + (−40 + 220i)10-s + 500·11-s − 288i·13-s − 16·14-s + 256·16-s + 1.51e3i·17-s + 1.34e3·19-s + (−880 − 160i)20-s + 2.00e3i·22-s + 4.10e3i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.983 + 0.178i)5-s + 0.0308i·7-s − 0.353i·8-s + (−0.126 + 0.695i)10-s + 1.24·11-s − 0.472i·13-s − 0.0218·14-s + 0.250·16-s + 1.27i·17-s + 0.854·19-s + (−0.491 − 0.0894i)20-s + 0.880i·22-s + 1.61i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.63276 + 1.36267i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.63276 + 1.36267i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-55 - 10i)T \) |
good | 7 | \( 1 - 4iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 500T + 1.61e5T^{2} \) |
| 13 | \( 1 + 288iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.51e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.34e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.10e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.28e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.89e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.40e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 8.90e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.98e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.59e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.08e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.46e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 2.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.69e4iT - 8.58e9T^{2} \) |
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
−13.59008462153963374369562770331, −12.59253517828722496480906700930, −11.15086976163081384741291349722, −9.800657460147906016716225429453, −9.040172510451602001855129483472, −7.58352100319190299151170309654, −6.32560384392382691006213829588, −5.43232744380374293338938650887, −3.64115891137828773068691009250, −1.48304210697544942514375858582,
1.02735312446622005614696693660, 2.50475634595175616871834590523, 4.25716412788135121132275490878, 5.68668026225452014553878987905, 7.08438496006970519398732240640, 9.005040534915891400101713237691, 9.477751498930544868759119109792, 10.77556339175268371557499246693, 11.85336586988649920030853644940, 12.81870899589157661423806784592