| L(s) = 1 | + (−9.88 − 9.88i)3-s + (−17.6 + 17.6i)5-s + 122. i·7-s − 47.4i·9-s + (301. − 301. i)11-s + (−384. − 384. i)13-s + 349.·15-s − 1.55e3·17-s + (2.09e3 + 2.09e3i)19-s + (1.20e3 − 1.20e3i)21-s − 4.76e3i·23-s − 625i·25-s + (−2.87e3 + 2.87e3i)27-s + (−1.90e3 − 1.90e3i)29-s − 1.15e3·31-s + ⋯ |
| L(s) = 1 | + (−0.634 − 0.634i)3-s + (−0.316 + 0.316i)5-s + 0.943i·7-s − 0.195i·9-s + (0.750 − 0.750i)11-s + (−0.630 − 0.630i)13-s + 0.401·15-s − 1.30·17-s + (1.33 + 1.33i)19-s + (0.598 − 0.598i)21-s − 1.87i·23-s − 0.200i·25-s + (−0.758 + 0.758i)27-s + (−0.421 − 0.421i)29-s − 0.215·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8224979980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8224979980\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (17.6 - 17.6i)T \) |
| good | 3 | \( 1 + (9.88 + 9.88i)T + 243iT^{2} \) |
| 7 | \( 1 - 122. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-301. + 301. i)T - 1.61e5iT^{2} \) |
| 13 | \( 1 + (384. + 384. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + 1.55e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-2.09e3 - 2.09e3i)T + 2.47e6iT^{2} \) |
| 23 | \( 1 + 4.76e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (1.90e3 + 1.90e3i)T + 2.05e7iT^{2} \) |
| 31 | \( 1 + 1.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (7.42e3 - 7.42e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 4.11e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (8.40e3 - 8.40e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 - 8.94e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-2.49e4 + 2.49e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + (2.32e4 - 2.32e4i)T - 7.14e8iT^{2} \) |
| 61 | \( 1 + (-1.43e4 - 1.43e4i)T + 8.44e8iT^{2} \) |
| 67 | \( 1 + (-3.88e4 - 3.88e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 1.28e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 2.85e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 5.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-3.94e4 - 3.94e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 9.72e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 4.67e4T + 8.58e9T^{2} \) |
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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
−11.26801540624394075868469626127, −10.12596891711467364914265853160, −8.991107689840635359705349078826, −8.127894524923778860779205915329, −6.89556251446022620645660666300, −6.19041679820874448881097568761, −5.26899788962534512775110606239, −3.72604345196042237614675970112, −2.45022699945539150155484202496, −0.931148690589104407927854643664,
0.29452088166631939204048223373, 1.83498674442448874752491554022, 3.73955553640778619261325072573, 4.58902847585057086906636361384, 5.34914484163564943243273960324, 7.01115819081835570148694612075, 7.39805838502602388161429070284, 9.095176970402657227005521259114, 9.629584131402815033387301720263, 10.78691718538773913738236443475
