L(s) = 1 | + 81i·3-s − 332. i·5-s − 3.96e3·7-s − 6.56e3·9-s − 3.23e3i·11-s + 1.15e5i·13-s + 2.69e4·15-s − 5.62e5·17-s − 1.04e6i·19-s − 3.21e5i·21-s − 1.24e6·23-s + 1.84e6·25-s − 5.31e5i·27-s + 1.09e6i·29-s + 8.23e6·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.238i·5-s − 0.624·7-s − 0.333·9-s − 0.0666i·11-s + 1.12i·13-s + 0.137·15-s − 1.63·17-s − 1.84i·19-s − 0.360i·21-s − 0.931·23-s + 0.943·25-s − 0.192i·27-s + 0.287i·29-s + 1.60·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.239275349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.239275349\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 81iT \) |
good | 5 | \( 1 + 332. iT - 1.95e6T^{2} \) |
| 7 | \( 1 + 3.96e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.23e3iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.15e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 5.62e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.04e6iT - 3.22e11T^{2} \) |
| 23 | \( 1 + 1.24e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.09e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 - 8.23e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.93e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 1.92e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.19e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 4.55e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.98e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 2.37e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 + 1.57e8iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 1.10e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 2.79e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 5.92e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.15e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.47e5iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 7.69e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.94e8T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.722129022462039657661418275697, −9.141695749384517779162804035701, −8.376022664960557245847188583310, −6.85791158785434298827291802022, −6.37626710119482342805299191937, −4.85633745100420659950222593770, −4.34041490095512152014286083097, −3.05569869715466078959751666775, −2.06386846417384631700448397468, −0.51734743020907516934071964455,
0.39785434427655394389408292742, 1.65258088374124635642849237308, 2.73446800213243560972148015914, 3.68198076169117847869990973446, 5.00704850019468352790421780884, 6.23016366037341001866520584965, 6.69301017890894715568710593080, 7.988235757859654605739957199018, 8.516282440429725678314544705341, 9.931240352584177041937676037084