| L(s) = 1 | − 1.41i·5-s − 2·7-s − 4.24i·11-s − 2.82i·13-s + 7.07i·17-s + (1 − 4.24i)19-s − 1.41i·23-s + 2.99·25-s + 10·29-s + 2.82i·31-s + 2.82i·35-s − 5.65i·37-s − 10·41-s − 12·43-s + 1.41i·47-s + ⋯ |
| L(s) = 1 | − 0.632i·5-s − 0.755·7-s − 1.27i·11-s − 0.784i·13-s + 1.71i·17-s + (0.229 − 0.973i)19-s − 0.294i·23-s + 0.599·25-s + 1.85·29-s + 0.508i·31-s + 0.478i·35-s − 0.929i·37-s − 1.56·41-s − 1.82·43-s + 0.206i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8375033265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8375033265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1 + 4.24i)T \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 7.07iT - 17T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 - 2.82iT - 31T^{2} \) |
| 37 | \( 1 + 5.65iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 - 1.41iT - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 14.1iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 11.3iT - 79T^{2} \) |
| 83 | \( 1 + 4.24iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 8.48iT - 97T^{2} \) |
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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
−8.414066668104710553780849846131, −8.097423073832669930039587754061, −6.68879116065859589580559759584, −6.32466854167710440351249863080, −5.36623782528449259751731534900, −4.65057334902231568529829007215, −3.44589469951260031632651024228, −2.97076155210468261316761445450, −1.41449687756811312408534331126, −0.27512453030734254081306282151,
1.53610189667986003881240686396, 2.71705027548671942655146982522, 3.35727701118365112113449531692, 4.55508378514584626340535146117, 5.11515380299465485510014742758, 6.51021849318124771537312908184, 6.71360108475616984799355836452, 7.47295660411565597160686734721, 8.388607560446760135603325460951, 9.347453729947505209716856214190
