| L(s) = 1 | + 0.585·5-s + 0.828i·7-s + 2.82i·11-s + 2.82i·13-s + 2.58i·17-s − 5.65·19-s − 6.82·23-s − 4.65·25-s − 3.41·29-s − 8.82i·31-s + 0.485i·35-s − 7.65i·37-s − 5.41i·41-s + 1.65·43-s − 4.48·47-s + ⋯ |
| L(s) = 1 | + 0.261·5-s + 0.313i·7-s + 0.852i·11-s + 0.784i·13-s + 0.627i·17-s − 1.29·19-s − 1.42·23-s − 0.931·25-s − 0.634·29-s − 1.58i·31-s + 0.0820i·35-s − 1.25i·37-s − 0.845i·41-s + 0.252·43-s − 0.654·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4647990727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4647990727\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 - 0.828iT - 7T^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 2.58iT - 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 + 8.82iT - 31T^{2} \) |
| 37 | \( 1 + 7.65iT - 37T^{2} \) |
| 41 | \( 1 + 5.41iT - 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 - 9.07T + 53T^{2} \) |
| 59 | \( 1 - 13.6iT - 59T^{2} \) |
| 61 | \( 1 - 3.65iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 10.4iT - 79T^{2} \) |
| 83 | \( 1 - 10.8iT - 83T^{2} \) |
| 89 | \( 1 + 3.75iT - 89T^{2} \) |
| 97 | \( 1 - 2.34T + 97T^{2} \) |
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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
−9.288809027281021738099627623361, −8.735089165159183120672806201126, −7.76175165049004360516421265597, −7.13723836688230777107165993107, −6.02824321065049013239422844552, −5.74321423432464136038343047358, −4.19161724723050316145218186929, −4.10046254065773747266944469205, −2.31309741102351129722389550244, −1.88315417137459952637792387924,
0.14521879525570596710742023155, 1.60036584838745865650294246436, 2.78637633829103657988950457794, 3.68172223563678435722019297893, 4.61679690207275194865865715882, 5.60954251541041735328909275286, 6.21334430589839571823284236589, 7.06976349748854356152661916998, 8.059508385014652739549220968848, 8.469876201794516617174647470508
